BusyBeaverWiki - User contributions [en]

1RB0LB 0RC0LD 1LD1RE 0LA0RF 0RD1RF 0RC---

2026-09-25T23:06:50Z

C7X: BMO redirect so it shows up on bbchallenge.org

#REDIRECT [[Beaver Math Olympiad]]

1RB0LD 0LC0RE 1LA0RD 0RB0LC 1RC1RF 0LD---

2026-09-25T23:06:18Z

C7X: BMO redirect so it shows up on bbchallenge.org

#REDIRECT [[Beaver Math Olympiad]]

1RB1LA 1RC0RD 1LA--- 1RE1RD 1LF0LA ---0LE

2026-09-25T23:05:28Z

C7X: BMO redirect

#REDIRECT [[Beaver Math Olympiad]]

BB(6)

2026-09-15T08:10:50Z

C7X: More champions

The 6-state, 2-symbol Busy Beaver problem, '''BB(6)''', refers to the unsolved 6<sup>th</sup> 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 ==
* In 1964, Green established Σ(5) ≥ 35.<ref name=":PMH">Pascal Michel. (last updated 2026). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm62 </ref>
* In 1972, Lynn established S(6) ≥ 522 and Σ(6) ≥ 42.<ref name=":PMH" />
* In 1983, Brady established S(6) ≥ 13,488 and Σ(6) ≥ 117.<ref name=":PMH" />
* In 1982, Schult established S(6) ≥ 4,208,824 and Σ(6) ≥ 2,075, although these bounds were only published later.<ref name=":PMH" />
* In 1990, Heiner Marxen and Jürgen Buntrock established first S(6) ≥ 13,122,572,797 and Σ(6) ≥ 136,612, then S(6) ≥ 8,690,333,381,690,951 and Σ(6) ≥ 95,524,079.<ref name=":PMH" />
* In July 2000, Heiner Marxen and Jürgen Buntrock established S(6) > 5.3 × 10<sup>42</sup> and Σ(6) > 2.5 × 10<sup>21</sup>.<ref name=":PMH" />
* In August 2000, Heiner Marxen and Jürgen Buntrock established first S(6) > 6.1 × 10<sup>119</sup> and Σ(6) > 1.4 × 10<sup>60</sup>, then S(6) > 6.1 × 10<sup>925</sup> and Σ(6) > 6.4 × 10<sup>462</sup>.<ref name=":PMH" />
* In February 2001, Heiner Marxen and Jürgen Buntrock established S(6) > 3.0 × 10<sup>1730</sup> and Σ(6) > 1.2 × 10<sup>865</sup>.<ref name=":PMH" />
* In November 2007, Terry and Shawn Ligocki established S(6) > 8.9 × 10<sup>1762</sup> and Σ(6) > 2.5 × 10<sup>881</sup>.<ref name=":PMH" />
* In December 2007, Terry and Shawn Ligocki established S(6) > 2.5 × 10<sup>2879</sup> and Σ(6) > 4.6 × 10<sup>1439</sup>.<ref name=":PMH" />
* In May 2010, Pavel Kropitz established S(6) > 3.8 × 10<sup>21132</sup> and Σ(6) > 3.1 × 10<sup>10566</sup>.<ref name=":PMH" />
* In June 2010, Pavel Kropitz established S(6) > 7.4 × 10<sup>36534</sup> and Σ(6) > 3.5 × 10<sup>18267</sup>.<ref name=":PMH" />
* In May 2022, Pavel Kropitz established S(6) > Σ(6) > 10&uparrow;&uparrow;15.<ref name=":PMH" />
* In June 2025, mxdys established S(6) > Σ(6) > 10&uparrow;&uparrow;11010000.<ref name=":PMH" />
* In June 2025, mxdys established S(6) > Σ(6) > 2&uparrow;&uparrow;2&uparrow;&uparrow;2&uparrow;&uparrow;10.<ref name=":PMH" />
[[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 1003 machines up to equivalence and 2190 machines not considering equivalence as of August 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 1101.

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
* {{TM|1RB0RF_1RC1RF_1LD0LE_---1LC_1RA1LE_1LC1RB|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}}
* {{TM|1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE|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)]]

BB(6)

2026-08-27T06:18:10Z

C7X: I guess "became known later" might be readable as either "discovered in computer output later" or "published later", the second one seems to be the situation

The 6-state, 2-symbol Busy Beaver problem, '''BB(6)''', refers to the unsolved 6<sup>th</sup> 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 ==
* In 1964, Green established Σ(5) ≥ 35.<ref name=":PMH">Pascal Michel. (last updated 2026). The Busy Beaver Competition: a historical survey. https://bbchallenge.org/~pascal.michel/ha#tm62 </ref>
* In 1972, Lynn established S(6) ≥ 522 and Σ(6) ≥ 42.<ref name=":PMH" />
* In 1983, Brady established S(6) ≥ 13,488 and Σ(6) ≥ 117.<ref name=":PMH" />
* In 1982, Schult established S(6) ≥ 4,208,824 and Σ(6) ≥ 2,075, although these bounds were only published later.<ref name=":PMH" />
* In 1990, Heiner Marxen and Jürgen Buntrock established first S(6) ≥ 13,122,572,797 and Σ(6) ≥ 136,612, then S(6) ≥ 8,690,333,381,690,951 and Σ(6) ≥ 95,524,079.<ref name=":PMH" />
* In July 2000, Heiner Marxen and Jürgen Buntrock established S(6) > 5.3 × 10<sup>42</sup> and Σ(6) > 2.5 × 10<sup>21</sup>.<ref name=":PMH" />
* In August 2000, Heiner Marxen and Jürgen Buntrock established first S(6) > 6.1 × 10<sup>119</sup> and Σ(6) > 1.4 × 10<sup>60</sup>, then S(6) > 6.1 × 10<sup>925</sup> and Σ(6) > 6.4 × 10<sup>462</sup>.<ref name=":PMH" />
* In February 2001, Heiner Marxen and Jürgen Buntrock established S(6) > 3.0 × 10<sup>1730</sup> and Σ(6) > 1.2 × 10<sup>865</sup>.<ref name=":PMH" />
* In November 2007, Terry and Shawn Ligocki established S(6) > 8.9 × 10<sup>1762</sup> and Σ(6) > 2.5 × 10<sup>881</sup>.<ref name=":PMH" />
* In December 2007, Terry and Shawn Ligocki established S(6) > 2.5 × 10<sup>2879</sup> and Σ(6) > 4.6 × 10<sup>1439</sup>.<ref name=":PMH" />
* In May 2010, Pavel Kropitz established S(6) > 3.8 × 10<sup>21132</sup> and Σ(6) > 3.1 × 10<sup>10566</sup>.<ref name=":PMH" />
* In June 2010, Pavel Kropitz established S(6) > 7.4 × 10<sup>36534</sup> and Σ(6) > 3.5 × 10<sup>18267</sup>.<ref name=":PMH" />
[[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 1016 machines up to equivalence and 2203 machines not considering equivalence as of August 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 1101.

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}}
* {{TM|1RB1RF_0LC0RF_1RD1LC_---0LE_0RC1LF_1RA0LE|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)]]

Fractran

2026-08-23T04:23:53Z

C7X: Formatting

'''Fractran''' (originally styled FRACTRAN) is an esoteric [[Turing complete]] model of computation invented by John Conway in 1987.<ref>Conway, John H. (1987). "FRACTRAN: A Simple Universal Programming Language for Arithmetic". ''Open Problems in Communication and Computation''. Springer-Verlag New York, Inc. pp. 4–26. <nowiki>http://doi.org/10.1007/978-1-4612-4808-8_2</nowiki></ref> In this model a program is simply a finite list of fractions (rational numbers), the program state is an integer. For more details see https://en.wikipedia.org/wiki/FRACTRAN.

Discord user Coda came up with a way to transform any Fractran program into a Turing Machine, see [https://discord.com/channels/960643023006490684/1438019511155691521/1441844795613122560 source].

'''BB_fractran'''(n) or '''BBf'''(n) is the Busy Beaver function for Fractran programs. Holdouts lists by Daniel Yuan: [https://github.com/int-y1/BBFractran/blob/main/holdout/README.md Holdouts lists]

== Definition ==
A Fractran program is a list of rational numbers <math>[q_0, q_1, \dots, q_{k-1}]</math> called rules and a Fractran state is an integer <math>s \in \mathbb{Z}</math>. The numerator and denominator of any rational number fraction do not share any prime factors (they are in reduced form). We say that a rule <math>q_i</math> applies to state <math>s</math> if <math>s \cdot q_i \in \mathbb{Z}</math>. If no rule applies, we say that the computation has halted otherwise we apply the first applicable rule at each step. In that case we say <math>s \to t</math> and <math>t = s \cdot q_i</math> and <math>i = \min \{ i : s \cdot q_i \in \mathbb{Z} \}</math>. As with [[Turing machines]], we will write <math>s \xrightarrow{N} t</math> if <math>s \to s_1 \to \cdots \to s_{N-1} \to t</math> (s goes to t after N steps) and <math>s \to^* t</math> or <math>s \to^+ t</math> if <math>s \xrightarrow{N} t</math> for some N≥0 or N≥1 (respectively). We say that a program has runtime N (or halts in N steps) starting in state s if <math>s \xrightarrow{N} t</math> and computation halts on t.

Let <math>\Omega(n)</math> be the total number of prime factors of a positive integer n. In other words, <math>\Omega(2^{a_0} 3^{a_1} \cdots p_n^{a_n}) = \sum_{k=0}^n a_k</math>. Then given a rule <math>\frac{a}{b}</math> we say that <math>\text{size} \left( \frac{a}{b} \right) = \Omega(a) + \Omega(b)</math>. And the size of a Fractran program <math>[q_0, q_1, \dots, q_{k-1}]</math> is <math>k + \sum_{i=0}^{k-1} \text{size}(q_i)</math>.

BB_fractran(n) or BBf(n) is the maximum runtime starting in state 2 for all halting Fractran programs of size n. It is a non-computable function akin to the [[Busy Beaver Functions]] since Fractran is Turing Complete.

== Vector Representation ==
Fractran programs are not easy to interpret, in fact it may be completely unclear at first that they can perform any computation at all. One of the key insights is to represent all numbers (states and rules) in their prime factorization form. For example, we can use a vector <math>[ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{Z}^n</math> to represent the number <math>2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math>.

Let the vector representation (for a sufficiently large n) for a state <math>a = 2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math> be <math>v(a) = [ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{N}^n</math> and the vector representation for a rule <math>\frac{a}{b}</math> be <math>v \left( \frac{a}{b} \right) = v(a) - v(b) \in \mathbb{Z}^n</math> (Note that this is just an extension of the original definition extended to allow negative <math>a_i</math>).

Now, rule q applies to state s iff <math>v(s) + v(q) \in \mathbb{N}^n</math> (all components of the vector are ≥0) and if <math>s \to t</math> then <math>v(t) = v(s) + v(q)</math>. So the Fractran multiplication model is completely equivalent to the vector adding model. For presentation, we will represent a Fractran program with a matrix where each row is the vector representation for a rule.

For example, the BBf(15) champion (<code>[1/45, 4/5, 3/2, 25/3]</code>) in vector representation would be:

<math display="block">\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>

In this representation, it becomes much easier to reason about Fractran programs and describe general rules. It is also very easy to calculate the size of a rule or program in vector representation. It is the sum of absolute values of all elements in the matrix + number of rules (number of rows).

=== Relationship to VAS / Petri Nets ===
Using vector representation, Fractran programs are a deterministic version of [[wikipedia:Vector_addition_system|Vector Addition Systems (VAS)]] (and, equivalently, [[wikipedia:Petri_net|Petri Nets]]). VAS are identical to Fractran programs in vector representation except that the rules are unordered and non-deterministic, they are used to model distributed systems where precise order of rule execution cannot be predicted. Interestingly, many problems about VAS are actually decidable, but their runtimes are extremely slow. Notably, the reachability problem (given states A and B are there a sequence of rules so that <math>A \to^* B</math>) is "Ackermann-complete" meaning that the optimal algorithm has worst-case runtime akin to the famously fast-growing Ackermann function.<ref>Czerwiński, Wojciech; Orlikowski, Łukasz (2021). ''Reachability in Vector Addition Systems is Ackermann-complete''. 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). https://arxiv.org/abs/2104.13866.</ref>

== Visualizing Fractran Programs' Space-Time Diagrams ==
Katelyn Doucette's Fractran space-time diagram visualizer produces the following space-time diagrams for some notable Fractran Programs, under the following principle: Each color represents a prime factor. Left -> right colors indicating the index of that register, and how wide the color is representing how big the value is at that step. Source code: https://github.com/Laturas/FractranVisualizer
{| class="wikitable"
|+
|[[File:Fractran_22_Cryptid.webp|alt=The space-time diagram of Fenrir|460x460px]]
The space-time diagram of Fenrir
|[[File:Hydra.webp|alt=The space-time diagram of Hydra.|460x460px]]
The space-time diagram of Hydra.
|[[File:Bbf21 champ full.png|alt=The space-time diagram of the BBf(21) champion.|400x400px]]

The space-time diagram of the BBf(21) champion. The width & height of the diagram can be set in the visualizer.
|[[File:Space_Needle.webp|alt=The space-time diagram of Space Needle.|460x460px]]
The space-time diagram of Space Needle.
|}

== Deciders ==
[[File:Fractran deciders.png|alt=Fractran deciders|thumb|All Fractran deciders summarized and their relations, shared by Daniel Yuan on [https://discord.com/channels/960643023006490684/1438019511155691521/1439001835904958655 14 Nov 2025]]]Many specialized deciders have been invented to prove Fractran programs non-halting. See image at right. There are three extra deciders: [https://discord.com/channels/960643023006490684/1438019511155691521/1449775657554022531 Spanning Vectors Masked,] which should be very effective, but implementing it is in-progress, a version of Spanning Vectors Masked - [https://discord.com/channels/960643023006490684/1438019511155691521/1453217977385091092 Masked Linear Invariant] - which is very powerful, and some holdouts were removed by [[User:Sligocki|Shawn Ligocki]] with [https://lsv.ens-paris-saclay.fr/Software/fast/ FAST] (Fast Acceleration of Symbolic Transition systems), a pre-existing general tool.

-d released a new decider on 25 Jan 2026: [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 Beeping Permutation].

TODO: create pages about the deciders.

== Champions ==
The table of champions is split into two pieces: the first for small champions (up to BBf(14)) which all share the same relatively simple behavior (sequential programs) is collapsed by default; the second for champions BBf(15) and beyond which have more complex and varied behavior.
All small champions as well as the first few larger ones were discovered and proven maximal by Jason Yuen (@-d) in their initial enumeration on [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025].

BBf(21) and below are solved. BBf(22) is the smallest domain to contain a Cryptid, and all other machines for BBf(22) are solved.

<div class="toccolours mw-collapsible mw-collapsed">'''Small Champions'''<div class="mw-collapsible-content">
{| 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>
|}
</div></div>

{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
!Champion Found
!Holdouts Proven
|-
| 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>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 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>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 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>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434313398799175710 1 Nov 2025]
|Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1362008236118511758/1434771877376557086 3 Nov 2025]
|-
| 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>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1435313806493614131 4 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1436661215911870584 8 Nov 2025]
|-
| 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>
|@creeperman7002 [https://discord.com/channels/960643023006490684/1362008236118511758/1435763150489387090 5 Nov 2025]
|Decider: Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438558242388312165 13 Nov 2025]
3 Holdouts: Racheline & Shawn Ligocki
|-
|20
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1438480761169776733 13 Nov 2025]
|Decider: Jason Yuen (@-d)
([https://github.com/int-y1/BBFractran/tree/main/holdout Enum+initial])
Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438559507579011194 13] and [https://discord.com/channels/960643023006490684/1438019511155691521/1438996636389998773 14 Nov 2025]

Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1447069110541484146 7] and [https://discord.com/channels/960643023006490684/1438019511155691521/1453213088630444168 24 Dec 2025]
6 Holdouts: Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1452913055053778945 23 Dec 2025]
|-
|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>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1439759182587891894 16 Nov 2025]
|No holdouts remain. Claude Opus 4.6's proof of nonhalting of all the 140 holdouts: [https://discord.com/channels/960643023006490684/1438019511155691521/1485168251997786173 28 March 2026]
|-
|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>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1448912286713384961 11 Dec 2025] and Jason Yuen (@-d)<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1448953682237460480 <nowiki>[1]</nowiki>]</sup>
|3 holdouts remain. Claude Opus 4.6 gave a proof of all machines but the 3 Fenrir Cryptids, see [https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 Discord].

The holdouts list whose elements are exactly the 3 Fenrir Cryptids on GitHub: [https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_3.txt sz22_3.txt]
Known [[Cryptid|Cryptids]]:

# Fenrir
|-
|23
|<math>> 4.393 \times 10^{124}</math>
|<code>[10/3, 9/14, 5/4, 121/2, 7/5, 3/11]</code>
|<math>\begin{bmatrix}
1 & -1 & 1 & 0 & 0 \\
-1 & 2 & 0 & -1 & 0 \\
-2 & 0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & -1 & 1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1510781736374763702 1 Jun 2026]
|21,295 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1511579969825013811 2 Jun 2026]
By August 5th, 2026, the unofficial holdouts count had been reduced to 13. [https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_13_unofficial.txt sz23_13_unofficial.txt]
Known [[Cryptid|Cryptids]]:

# 11 Hydra-like Cryptids (including Frankenstein's Monster and Antihydra-like Cryptid)
|-
|24
|<math>> 9.263 \times 10^{9595}</math>
|<code>[18/35, 1/10, 11/5, 75/2, 49/3, 5/11]</code>
|<math>\begin{bmatrix}
1 & 2 & -1 & -1 & 0 \\
-1 & 0 & -1 & 0 & 0 \\
0 & 0 & -1 & 0 & 1 \\
-1 & 1 & 2 & 0 & 0 \\
0 & -1 & 0 & 2 & 0 \\
0 & 0 & 1 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1540835921182728272 22 Aug 2026]
|No holdouts list yet.

An informal list from a 20%-complete enumeration: [https://discord.com/channels/960643023006490684/1438019511155691521/1540831684071530637 22 Aug 2026]
|}

=== Behavior of Champions ===

==== Sequential programs ====
All champions up to BBf(14) have very simple behavior. They are all of the form: <math>\left[ \frac{3^{a_1}}{2}, \frac{5^{a_2}}{3}, \dots, \frac{p_n^{a_k}}{p_{k-1}}, \frac{1}{p_k} \right]</math> or in vector representation (limited to k=4):

<math display="block">\begin{bmatrix}
-1 & a_1 & 0 & 0 & 0 \\
0 & -1 & a_2 & 0 & 0 \\
0 & 0 & -1 & a_3 & 0 \\
0 & 0 & 0 & -1 & a_4 \\
0 & 0 & 0 & 0 & -1
\end{bmatrix}</math>

These champions repeatedly apply the rules in sequence, never going back to a previous rule. They apply the first rule until they've exhausted all 2s, then the second rule until they've exhausted all 3s, etc. They have a runtime of <math>1 + a_1 + a_1 a_2 + a_1 a_2 a_3 + \cdots = \sum_{i=0}^k \prod_{j=1}^i a_j</math> and size <math>2k+2 + \sum_{i=1}^k a_i</math>. This grows linearly for k=1 (BBf(5) to BBf(10)) and quadratically for k=2 (BBf(11) to BBf(14)). Letting k grow with the size, the maximum runtime grows exponentially in the program size.

==== BBf(15) Family ====
The BBf(15) and BBf(16) champions are members of a family of programs (parameterized by <math>n \ge 1</math>):

<math display="block">\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & n
\end{bmatrix}</math>

Let a = 2, b = 3, and c = 5.

The BBf(15) champion (n = 2) implements this iteration:

<math display="block">\begin{array}{lcl}
b^0 & \xrightarrow{0} & \text{halt} \\
b^1 & \xrightarrow{7} & b^4 \\
b^2 & \xrightarrow{7} & b^5 \\
b^3 & \xrightarrow{5} & b^2 \\
b^4 & \xrightarrow{5} & b^3 \\
b^{k+5} & \xrightarrow{3} & b^k \\
\end{array}</math>

which follows a permutation-like trajectory: <math>a \xrightarrow{1} b^1 \to b^4 \to b^3 \to b^2 \to b^5 \to b^0 \to \text{halt}</math>

The BBf(16) champion (n = 3) implements this iteration:

<math display="block">\begin{array}{lcl}
b^0 & \xrightarrow{0} & \text{halt} \\
b^1 & \xrightarrow{10} & b^6 \\
b^2 & \xrightarrow{10} & b^7 \\
b^3 & \xrightarrow{8} & b^4 \\
b^4 & \xrightarrow{8} & b^5 \\
b^5 & \xrightarrow{6} & b^2 \\
b^6 & \xrightarrow{6} & b^3 \\
b^{k+7} & \xrightarrow{4} & b^k \\
\end{array}</math>

which follows a permutation-like trajectory: <math>a \xrightarrow{1} b^1 \to b^6 \to b^3 \to b^4 \to b^5 \to b^2 \to b^7 \to b^0 \to \text{halt}</math>

==== BBf(17) Family ====
The BBf(17) to BBf(19) champions are members of a family of programs (parameterized by <math>m,n \ge 0</math>)

<math display="block">\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & n \\
0 & 1 & -1 & 0 \\
m & 0 & 1 & -1
\end{bmatrix}</math>

which have size <math>m+n+12</math>

This family obeys the following rules:

# <math>[1, 0, 0, 0] \xrightarrow{1} [0, 0, 0, n]</math>
# if d≥1 and b≤m:<math display="block">[0, b, 0, d] \xrightarrow{m+b+2} [0, b+1, 0, d - 1 + n(m-b)]</math>
# if d≥1 and b≥m:<math display="block">[0, b, 0, d] \xrightarrow{2m+2} [0, b+1, 0, d - 1]</math>
#if d=0: [0,b,0,d] has halted

and furthermore these rules are applied in order since b is always increasing (and d is eventually decreasing). Combining these together we get runtime:<math display="block">1 + n(m+1)(m(m+1)+2) - \frac{m(m+1)}{2}</math>

The optimal choices for n,m for various program sizes are:
{| class="wikitable"
|+
!Size
!n
!m
!Runtime
|-
|16
|1
|3
|51
|-
|'''''17'''''
|'''''2'''''
|'''''3'''''
|'''''107'''''
|-
|'''''18'''''
|'''''2'''''
|'''''4'''''
|'''''211'''''
|-
|'''''19'''''
|'''''2'''''
|'''''5'''''
|'''''370'''''
|-
|20
|2
|6
|596
|-
|21
|3
|6
|904
|}

<br>
==== BBf(20) ====
[[File:Screenshot 2026-04-01 104704.png|alt=Full space-time diagram of the BBf(20) champion.|left|507x507px]]
The BBf(20) champion (running 746 steps):

<math display="block">\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>

This program implements a [[Collatz-like]] iteration. Let <math>C(n) = [0, 0, n, 2, 0]</math>, then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{49} & C(2) \\
C(3k) & \xrightarrow{3k} & \text{halt} \\
C(3k+1) & \xrightarrow{11k+22} & C(4k+3) \\
C(3k+2) & \xrightarrow{11k+22} & C(4k+4) \\
\end{array}</math>

which follows the reasonably "lucky" trajectory:

<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>

==== BBf(21) ====
[[File:Bbf21 champ full.png|alt=The full space-time diagram of the BBf(21) champion until halting.|thumb|The full space-time diagram of the BBf(21) champion until halting.]]

The BBf(21) champion (running >31M steps):
<math display="block">\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>

This program implements a Collatz-like iteration. Let <math>D(n) = [0, 0, n, 0]</math>, then:<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1439779341365022852]</sup>
<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & D(1) \\
D(3k) & \xrightarrow{k} & \text{halt} \\
D(3k+1) & \xrightarrow{21k+7} & C(10k+4) \\
D(3k+2) & \xrightarrow{21k+14} & C(10k+7) \\
\end{array}</math>

which follows the reasonably "lucky" trajectory:
<math display="block">\begin{array}{ll}
D(1) & \to D(4) \to D(14) \to D(47) \to D(157) \to D(524) \to D(1747) \to D(5824) \to D(19414) \\
& \to D(64714) \to D(215714) \to D(719047) \to D(2396824) \to D(7989414) \to \text{halt} \\
\end{array}</math>

==== BBf(22) ====
The BBf(22) champion (running <math>> 10^{62}</math> steps):
<math display="block">\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>

This program implements a [[Collatz-like]] unbiased pseudo-random walk. Let <math>S(x,y) = [0, 0, x, 0, y]</math>, then:<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1449118888142049421]</sup>
<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & S(0,1) \\
S(x, 0) & = & \text{halt} \\
S(3k, y+1) & \xrightarrow{14k+4} & S(5k+1, y+1) \\
S(3k+1, y+1) & \xrightarrow{14k+10} & S(5k+3, y+2) \\
S(3k+2, y+1) & \xrightarrow{14k+12} & S(5k+4, y) \\
\end{array}</math>

This pseudo-random walk iterates 275 times until it halts reaching a maximum y value of 14 at iteration 111:
<math display="block">\begin{array}{ll}
S(0,1) & \to S(1,1) \to S(3,2) \to S(6,2) \to S(11, 2) \to S(19, 1) \to S(33, 2) \to S(56, 2) \to S(94, 1) \\
& \to S(158, 2) \to S(264, 1) \to S(441, 1) \to S(736, 1) \to S(1228, 2) \to S(2048, 3) \\
& \vdots \\
& \to S(4065328691604230522442358, 13) \\
& \to S(6775547819340384204070598, 14) \\
& \to S(11292579698900640340117664, 13) \\
& \vdots \\
& \to S(27930059557111373800280446055462487109112535227834136644, 2) \\
& \to S(46550099261852289667134076759104145181854225379723561074, 1) \\
& \to S(77583498769753816111890127931840241969757042299539268458, 2) \\
& \to S(129305831282923026853150213219733736616261737165898780764, 1) \\
& \to S(215509718804871711421917022032889561027102895276497967941, 1) \\
& \to S(359182864674786185703195036721482601711838158794163279903, 2) \\
& \vdots \\
& \to S(5894430516013404355095519889620117404469367857588232386361874, 2) \\
& \to S(9824050860022340591825866482700195674115613095980387310603124, 1) \\
& \to S(16373418100037234319709777471166992790192688493300645517671874, 0)
\end{array}</math>
If it were a truly random walk, there would be a 5.9% chance that it takes at least 275 steps to reach 0. So this program is mildly lucky.
==== BBf(23) ====
The BBf(23) champion
<math>\begin{bmatrix}
1 & -1 & 1 & 0 & 0 \\
-1 & 2 & 0 & -1 & 0 \\
-2 & 0 & 1 & 0 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & -1 & 1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math> follows an unbiased [[Collatz-like]] pseudo-random walk:
let <math>A(x,y)=[0,0,0,x,y]</math>

<math>[1,0,0,0,0] \to A(0,2)</math>

<math>A(2x,y) \xrightarrow{12x+4} A(5x+1,y+1)</math> if y>0

<math>A(2x+1,y) \xrightarrow{12x+10} A(5x+4,y-1)</math> if y>0

<math>A(x,0) \to halt</math>

<math>A(0,2) \to A(1,3) \to A(4,2) \to A(11,3) \to A(29,2) \to A(74,1) \to A(186,2) \to A(466,3) \to A(1166,4) \to A(2916,5) \to A(7291,6) \to \dots</math>

== Cryptids ==

=== Size 22: Fenrir ===
[[File:Fractran 22 Cryptid.webp|alt=The space-time diagram of Fenrir.|thumb|Partial space-time diagram of Fenrir.]]
"Fenrir" is a family of 3 size 22 [[Cryptids]] discovered by Jason Yuen (@-d) and Claude Opus 4.6 on 22 Mar 2026. Out of 2003 holdouts of size 22, Claude Opus 4.6 used Lean to prove that 1997 holdouts were non-halting and 3 holdouts were halting. The remaining 3 holdouts are the Fenrir family.<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1485415054475268179]</sup> Discord user @ZTS439 shared [https://discord.com/channels/960643023006490684/1438019511155691521/1487251919444508723 some analysis] and a [https://discord.com/channels/960643023006490684/1438019511155691521/1487252789158613002 Python program] for it. Its name comes from [[wikipedia:Norse_mythology|nordic mythology]]; [[wikipedia:Fenrir|Fenrir]] is the wolf that helps destroy the world during [[wikipedia:Ragnarök|Ragnarök]].

{| class="wikitable"
|+
!Holdout number
!Holdout
!Vector Representation
|-
| 29/2003
| <code>[1/15, 27/77, 49/3, 10/49, 33/2]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 0 & 0 \\
0 & 3 & 0 & -1 & -1 \\
0 & -1 & 0 & 2 & 0 \\
1 & 0 & 1 & -2 & 0 \\
-1 & 1 & 0 & 0 & 1
\end{bmatrix}</math>
|-
| 41/2003
| <code>[1/15, 49/3, 27/77, 10/49, 33/2]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 0 & 0 \\
0 & -1 & 0 & 2 & 0 \\
0 & 3 & 0 & -1 & -1 \\
1 & 0 & 1 & -2 & 0 \\
-1 & 1 & 0 & 0 & 1
\end{bmatrix}</math>
|-
| 430/2003
| <code>[27/35, 1/33, 25/3, 22/25, 21/2]</code>
|<math>\begin{bmatrix}
0 & 3 & -1 & -1 & 0 \\
0 & -1 & 0 & 0 & -1 \\
0 & -1 & 2 & 0 & 0 \\
1 & 0 & -2 & 0 & 1 \\
-1 & 1 & 0 & 1 & 0
\end{bmatrix}</math>
|}

All 3 holdouts follow a biased random walk that somewhat resembles [[Hydra]]. Let <math>S(x,y) = [x, 0, 0, 2, y]</math> (for 29/2003 and 41/2003) or <math>S(x,y) = [x, 0, 2, y, 0]</math> (for 430/2003), then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \to & S(0,1) \\
S(0, 2y) & = & \text{halt} \\
S(x, 2y) & \to & S(x-1, 5y+2) \\
S(x, 2y+1) & \to & S(x+2, 5y)
\end{array}</math>

The first few visited states are $$S(0, 1) \to S(2, 0) \to S(1, 2) \to S(0, 7) \to S(2, 15) \to S(4, 35)$$

=== Size 23: 11 Hydra-like Cryptids ===

Define <math>Hydra(r_\mathrm{num}, r_\mathrm{den}, x_\mathrm{offset}, y_\mathrm{offset}, (x_\mathrm{init}, y_\mathrm{init}))</math> to be the problem as follows:

# The initial state is <math>(x_\mathrm{init}, y_\mathrm{init})</math>.
# The iteration <math>(x, y) \mapsto (r_\mathrm{num} \times \lfloor x/r_\mathrm{den} \rfloor + x_\mathrm{offset}[x \bmod r_\mathrm{den}], y + y_\mathrm{offset}[x \bmod r_{den}])</math> is repeated. Here, <math>x_\mathrm{offset}, y_\mathrm{offset}</math> are 0-indexed.
# Are all the values of y non-negative?

Furthermore, the Hydra-like problem is considered a [[Cryptids|Cryptid]] if it also satisfies:

# <math>y_\mathrm{offset}</math> contains a negative number, and the average is positive.
# <math>x \bmod r_\mathrm{den}</math> is a pseudorandom sequence.
#There are no negative values of <math>y</math> early on.

For example, Fenrir is non-halting if and only if Hydra(5, 2, [2, 0], [-1, 2], (1, 0)). In this Hydra problem, the first few visited states are <math>(1, 0) \to (0, 2) \to (2, 1) \to (7, 0) \to (15, 2) \to (35, 4)</math>.

There are 11 Hydra-like Cryptids of size exactly 23, listed in the table below. All 11 Cryptids are not correlated with each other. Fenrir is included in the table as a reference.
{| class="wikitable sortable"
|+
!Domain: Holdout Number
!Holdout
!Hydra problem
!Notes
|-
|BBf(23): #11/694
|<code>[1/135, 25/21, 33/5, 2/3, 7/11, 5/2]</code>
|Hydra(7, 4, [1, 3, 5, 6], [0, 1, 2, -1], (1, 0))
|
|-
|BBf(22): #29/2003
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code>
|Hydra(5, 2, [2, 0], [-1, 2], (1, 0))
|Fenrir
|-
|BBf(23): #26/694
|<code>[1/15, 49/3, 81/77, 10/49, 33/2]</code>
|Hydra(7, 2, [-1, 8], [3, -1], (1, 2))
|
|-
|BBf(23): #47/694
|<code>[1/18, 4/15, 21/2, 121/3, 5/7, 2/11]</code>
|Hydra(5, 3, [1, 3, 4], [1, 3, -1], (1, 1))
|If #47 doesn't halt then Frankenstein's Monster doesn't halt
|-
|BBf(23): #77/694
|<code>[1/54, 4/15, 21/2, 11/3, 5/7, 3/11]</code>
|Hydra(7, 4, [0, 2, 4, 5], [0, 1, 2, -1], (1, 0))
|Same ratio 7/4 as #11
|-
|BBf(23): #151/694
|<code>[14/15, 1/12, 11/3, 63/2, 5/7, 2/11]</code>
|Hydra(8, 5, [1, 2, 4, 6, 7], [2, -1, 1, 3, 0], (0, 1))
|
|-
|BBf(23): #159/694
|<code>[14/15, 1/6, 121/3, 63/2, 5/7, 2/11]</code>
|Hydra(5, 3, [1, 2, 4], [3, -1, 1], (1, 3))
|Same ratio 5/3 as #47
|-
|BBf(23): #207/694
|<code>[2/15, 1/12, 441/2, 11/3, 5/7, 2/11]</code>
|Hydra(6, 5, [1, 3, 3, 5, 5], [1, 3, 0, 2, -1], (1, 1))
|
|-
|BBf(23): #218/694
|<code>[2/15, 1/6, 441/2, 121/3, 5/7, 2/11]</code>
|Hydra(4, 3, [1, 2, 3], [3, 1, -1], (1, 3))
|
|-
|BBf(23): #317/694
|<code>[4/15, 1/18, 63/2, 11/3, 5/7, 2/11]</code>
|Hydra(5, 4, [1, 2, 3, 5], [1, 0, -1, 2], (1, 1))
|
|-
|BBf(23): #319/694
|<code>[4/15, 1/24, 21/2, 11/3, 5/7, 3/11]</code>
|Hydra(5, 4, [0, 2, 4, 3], [0, 1, 2, -1], (1, 0))
|Same ratio 5/4 as #317
|-
|BBf(23): #323/694
|<code>[4/15, 1/6, 21/2, 1331/3, 5/7, 2/11]</code>
|Hydra(3, 2, [1, 2], [2, -1], (1, 2))
|If #323 doesn't halt then Antihydra-like Cryptid doesn't halt
|}

As of August 2026, the 13 unofficial holdouts are as follows: Fenrir, 11 Hydra-like Cryptids of size 23, and #601 <code>[9/10, 1/42, 22/3, 49/2, 5/11, 3/7]</code>.

==== Frankenstein's Monster ====
[[File:Frankenstein's Monster.webp|alt=Partial space-time diagram of Frankenstein's Monster.|thumb|Partial space-time diagram of Frankenstein's Monster.]]
"Frankenstein's Monster" is a size 23 [[Cryptid]]. It was created by tweaking a single instruction in the size 22 champion. This tweak switches it from a unbiased random walk to a biased one and thus makes halting probviously impossible. It is called Frankenstein's Monster since it was found by a combination of exhaustive search and hand design.<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1449138938215141478]</sup>

<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code> <math display="block">\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>

Its behavior is extremely similar to the size 22 champion. Let <math>S(x,y) = [0, 0, x, 0, y]</math>, then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & S(0,2) \\
S(x, 0) & = & \text{halt} \\
S(3k, y+1) & \xrightarrow{14k+4} & S(5k+1, y+2) \\
S(3k+1, y+1) & \xrightarrow{14k+10} & S(5k+3, y+4) \\
S(3k+2, y+1) & \xrightarrow{14k+12} & S(5k+4, y)
\end{array}</math>

with the only difference that the <math>y</math> values now change by <math>\{+1,+3,-1\}</math> depending on the value of <math>x\bmod 3</math> (instead of <math>\{0,+1,-1\}</math> in the original size 22 program). The <math>x</math> values follow the exact same path as in the original size 22 champion, but the y values quickly grow linearly with the number of iterations (as expected by the random model):
0: S(0, 1) @ 1 (0.00s)
100_000: S(10^22_185, 100171) @ 10^22_186 (0.87s)
200_000: S(10^44_370, 200187) @ 10^44_371 (3.42s)
300_000: S(10^66_555, 300759) @ 10^66_556 (7.68s)
400_000: S(10^88_740, 400451) @ 10^88_741 (13.64s)
500_000: S(10^110_925, 500421) @ 10^110_925 (21.28s)
600_000: S(10^133_109, 600351) @ 10^133_110 (30.62s)
700_000: S(10^155_294, 700319) @ 10^155_295 (41.64s)
800_000: S(10^177_479, 799911) @ 10^177_480 (54.30s)
900_000: S(10^199_664, 900259) @ 10^199_665 (68.59s)
1_000_000: S(10^221_849, 1000853) @ 10^221_850 (84.51s)
...
4_000_000: S(10^887_395, 4000201) @ 10^887_396 (1474.02s)
...
27_500_000: S(10^6_100_841, 27512703) @ 10^6_100_842 (87616.45s)

==== Antihydra-like Cryptid ====
This Cryptid is a size 23 [[Cryptid]]. This Cryptid was [https://discord.com/channels/960643023006490684/1438019511155691521/1449293536737361973 constructed by Maksandchael] by tweaking Frankenstein's Monster to make it as similar to [[Antihydra]] as possible. <code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>

<math display="block">\begin{bmatrix}
-1 & 2 & -1 & 0 & 0 \\
-1 & -1 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 3 \\
1 & -1 & 0 & 1 & 0 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math><pre>
H(a, b) = [0, 0, a-2, 0, b]
Start -> H(2, 3)
H(2a, b) -> H(3a, b+2)
H(2a+1, b+1) -> H(3a+1, b)
H(a,0) -> halt
</pre>

=== Size 25: Hydra ===
[[File:Hydra.webp|alt=Partial space-time diagram of Hydra.|thumb|300x300px|Partial space-time diagram of Hydra.]]
A size 25 program was produced and golfed by hand to simulate [[Hydra]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1449829146040467681 Discord]):

<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
<math display="block">
\begin{bmatrix}
-1 & 1 & 0 & -1 & 2 \\
-1 & 0 & 3 & 0 & 0 \\
1 & -1 & 0 & -1 & 1 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & -1 \\
1 & 0 & -1 & 1 & 0
\end{bmatrix}
</math>

The intended interpretation is that if we let <math>S(h,w) = [1, 0, w, h-3, 0]
</math> then it follows the following rules:

<math display="block">\begin{array}{lcl}
[1,0,\dots] & = & S(3, 0) \\
S(2k, 0) & \to^* & \text{halt} \\
S(2k, w+1) & \to^* & S(3k, w) \\
S(2k+1, w) & \to^* & S(3k+1, w+2)
\end{array}</math>

=== Size 36: BMO1 ===
[[File:Ftran bmo1.png|alt=Partial space-time diagram of BMO 1.|thumb|Partial space-time diagram of BMO 1.]]
A size 36 program was produced by hand to simulate [[BMO1]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1440018895212642424 Discord]):

<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
<math display="block">
\begin{bmatrix}
0 & 2 & -1 & 0 & -1 & 0 & 1 \\
1 & 0 & 0 & 0 & -1 & 0 & 0 \\
1 & 0 & -1 & -1 & 0 & 1 & 0 \\
0 & 1 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & -1 \\
0 & 0 & 0 & 1 & 0 & -1 & 0 \\
-1 & -1 & 2 & 0 & 0 & 0 & 0 \\
-1 & 0 & 1 & 0 & 1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 & 0 & 0
\end{bmatrix}
</math>

Let <math>A(a,b) = [a, b, 0, 0, 0, 0, 0]</math>, then it follows the rules:

<math display="block">\begin{array}{lcl}
[1,0,\dots] & \to^* & A(1, 2) \\
A(a, b) & \to^* & A(a-b, 4b+2) & \text{if } a > b \\
A(a, b) & \to^* & A(2a+1, b-a) & \text{if } a < b \\
A(a, b) & \to^* & \text{Halt} & \text{if } a = b
\end{array}</math>

=== Size 48: BMO 6 (“Space Needle”) ===
[[File:Space Needle.webp|alt=Partial space-time diagram of Space Needle.|thumb|Partial space-time diagram of Space Needle.]]
A size 48 program was produced by hand to simulate [https://wiki.bbchallenge.org/wiki/1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD BMO 6] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1441137371046482071 Discord])

<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>

<math>\begin{bmatrix}
-1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & -1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 \\
0 & 1 & 1 & -1 & 0 & 0 & -1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\
0 & -1 & 3 & 0 & 1 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 5 & 0 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & -1 & 0 & 0
\end{bmatrix}</math><pre>A(a, b) = B^a C^b E or B^(a-2) C^b D E

Start: A(7, 1)

A(1, b) --> halt

A(2a, b) --> A(5a+b+2, 1)

A(2a+1, b) --> A(b-1, b+c+3)</pre>

== References ==
<references />

[[Category:Functions]]

Diophantine Equation

2026-08-19T04:09:12Z

C7X: Ah you're right, thanks. Hopefully this one works, it seems to have been missed in Grechuk's preprint

A '''Diophantine equation''' is a polynomial equation with integer coefficients for which only integer solutions are sought. Diophantine equations are known to be Turing complete via the [https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's_theorem MRDP theorem].

== Definitions ==
A Diophantine equation is defined by a multi-variate polynomial <code>P(x,y,z,...)</code>. Solutions are values x,y,z,... such that <code>P(x,y,z,...) = 0</code>. The "magnitude" of a solution is max(|x|, |y|, |z|, ...) (the maximum absolute value of all it's variable assignments). The "score" for a Diophantine equation is the minimum magnitude across all solutions. In other words, it is the minimum N such that there exists a solution with -N ≤ x,y,z,... ≤ N.

The "size" or "height" of a Diophantine equation is the sum of <code>|coefficient|*2^degree</code> among all of its terms. For example <math>x^2 + xy + 6y + 1 = 0</math> has size <math>1 \cdot 2^2 + 1 \cdot 2^2 + 6 \cdot 2^1 + 1 \cdot 2^0 = 21 </math>.

<code>BBdio(H)</code> is the maximum score among all solvable polynomial Diophantine equations of height <code>H</code>.

== Champions ==
For 2 ≤ n ≤ 15: <code>BBdio(n) = n-2</code> via the trivial equation <math>x - (n-2) = 0</math>.[https://discord.com/channels/960643023006490684/1461768250575687897/1461769847754064036]
{| class="wikitable"
|+
!n
!BBdio(n)
!Champion
!Min Solution
!Source
|-
|2 ≤ n ≤ 20
|≥ n-2
|<math>x - (n-2) = 0</math>
|(n-2)
|
|-
|21
|≥ 26
|<math>x^2 + xy + 6y + 1 = 0</math>
|(-5, -26)
|
|-
|22
|≥ 22
|<math>x^3 + y^2 + 4 = 3x</math>
|(-8, ±22)
|
|-
|23
|≥ 31
|<math>x^2 + xy + x + 1 = 6y</math>
|(5, -31)
|
|-
|24
|≥ 66
|<math>x^3 + xy + 5y = 2</math>
|(-4, -66)
|
|-
|25
|≥ 470
|<math>x^3 + xy + y^2 + y + 7 = 0</math>
|(-63, -470)
|
|-
|26
|≥ 40
|<math>x^2 + xy + 7y + 4 = 0</math>
|
|
|-
|27
|≥ 849
|<math>x^3 + x^2 + y^2 + 9 = x</math>
|
|
|-
|28
|≥ 74
|<math>x^3 + xy + 2x + 5y = 2</math>
|
|
|-
|29
|> 10<sup>6</sup>
|<math>x^3 + x y^2 + x + z^3 - z + 1 = 0</math>
|(-4280795, 4360815, 5427173)*
|Andrew R. Booker via https://arxiv.org/abs/2108.08705 v3 page 23
|-
|...
|
|
|
|
|-
|33
|> 10<sup>11</sup>
|<math>x^4 + y^2z + z^2 - z + 3 = 0</math>
|(708313, 1099536, -267298981516)*
|https://arxiv.org/abs/2108.08705 v3 page 27
|-
|...
|
|
|
|
|-
|57
|> 10<sup>15</sup>
|<math>x^3 + y^3 + z^3 = 33</math>
|(-2736111468807040, -8778405442862239, 8866128975287528)*
|https://oeis.org/A060467
|-
|66
|> 10<sup>16</sup>
|<math>x^3 + y^3 + z^3 = 42</math>
|(12602123297335631, 80435758145817515, −80538738812075974)*
|https://oeis.org/A060467
|}
Note: For large champions (marked with * in Min Solution column) these are not known to be strictly the smallest magnitude solutions, instead they appear to be solutions with the minimal smallest value. If that is the case, then this still bounds BBdio since the score of these equations must be ≥ the minimal smallest variable value.

== Cryptids ==
There are several open problems in the [[wikipedia:Sums_of_three_cubes|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.

== See Also ==

* [https://discord.com/channels/960643023006490684/1461768250575687897 Discord BBdio thread]

[[Category:Functions]]

Diophantine Equation

2026-08-19T03:55:47Z

C7X: Please remove if this is not rigorous/well-sourced enough of an edit

A '''Diophantine equation''' is a polynomial equation with integer coefficients for which only integer solutions are sought. Diophantine equations are known to be Turing complete via the [https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's_theorem MRDP theorem].

== Definitions ==
A Diophantine equation is defined by a multi-variate polynomial <code>P(x,y,z,...)</code>. Solutions are values x,y,z,... such that <code>P(x,y,z,...) = 0</code>. The "magnitude" of a solution is max(|x|, |y|, |z|, ...) (the maximum absolute value of all it's variable assignments). The "score" for a Diophantine equation is the minimum magnitude across all solutions. In other words, it is the minimum N such that there exists a solution with -N ≤ x,y,z,... ≤ N.

The "size" or "height" of a Diophantine equation is the sum of <code>|coefficient|*2^degree</code> among all of its terms. For example <math>x^2 + xy + 6y + 1 = 0</math> has size <math>1 \cdot 2^2 + 1 \cdot 2^2 + 6 \cdot 2^1 + 1 \cdot 2^0 = 21 </math>.

<code>BBdio(H)</code> is the maximum score among all solvable polynomial Diophantine equations of height <code>H</code>.

== Champions ==
For 2 ≤ n ≤ 15: <code>BBdio(n) = n-2</code> via the trivial equation <math>x - (n-2) = 0</math>.[https://discord.com/channels/960643023006490684/1461768250575687897/1461769847754064036]
{| class="wikitable"
|+
!n
!BBdio(n)
!Champion
!Min Solution
!Source
|-
|2 ≤ n ≤ 20
|≥ n-2
|<math>x - (n-2) = 0</math>
|(n-2)
|
|-
|21
|≥ 26
|<math>x^2 + xy + 6y + 1 = 0</math>
|(-5, -26)
|
|-
|22
|≥ 22
|<math>x^3 + y^2 + 4 = 3x</math>
|(-8, ±22)
|
|-
|23
|≥ 31
|<math>x^2 + xy + x + 1 = 6y</math>
|(5, -31)
|
|-
|24
|≥ 66
|<math>x^3 + xy + 5y = 2</math>
|(-4, -66)
|
|-
|25
|≥ 470
|<math>x^3 + xy + y^2 + y + 7 = 0</math>
|(-63, -470)
|
|-
|26
|≥ 40
|<math>x^2 + xy + 7y + 4 = 0</math>
|
|
|-
|27
|≥ 849
|<math>x^3 + x^2 + y^2 + 9 = x</math>
|
|
|-
|28
|≥ 74
|<math>x^3 + xy + 2x + 5y = 2</math>
|
|
|-
|29
|> 10<sup>6</sup>
|<math>x^3 + x y^2 + x + z^3 - z + 1 = 0</math>
|(-4280795, 4360815, 5427173)*
|Andrew R. Booker via https://arxiv.org/abs/2108.08705 v3 page 23
|-
|...
|
|
|
|
|-
|48
|> 10<sup>10</sup>
|<math>x^3 + y^3 + z^3 = 24</math>
|(-2901096694, -15550555555, 15584139827)*
|https://oeis.org/A060467
|-
|57
|> 10<sup>15</sup>
|<math>x^3 + y^3 + z^3 = 33</math>
|(-2736111468807040, -8778405442862239, 8866128975287528)*
|https://oeis.org/A060467
|-
|66
|> 10<sup>16</sup>
|<math>x^3 + y^3 + z^3 = 42</math>
|(12602123297335631, 80435758145817515, −80538738812075974)*
|https://oeis.org/A060467
|-
|352
|> 10<sup>80</sup>
|<math>\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y} = 4</math>
|(<math> >4.373\times 10^{78}</math>, <math> >3.687\times 10^{79}</math>, <math> >1.544\times 10^{80}</math>)
|https://oeis.org/A369896
|}
Note: For large champions (marked with * in Min Solution column) these are not known to be strictly the smallest magnitude solutions, instead they appear to be solutions with the minimal smallest value. If that is the case, then this still bounds BBdio since the score of these equations must be ≥ the minimal smallest variable value.

== Cryptids ==
There are several open problems in the [[wikipedia:Sums_of_three_cubes|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.

== See Also ==

* [https://discord.com/channels/960643023006490684/1461768250575687897 Discord BBdio thread]

[[Category:Functions]]

Diophantine Equation

2026-08-19T02:21:47Z

C7X:

A '''Diophantine equation''' is a polynomial equation with integer coefficients for which only integer solutions are sought. Diophantine equations are known to be Turing complete via the [https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich's_theorem MRDP theorem].

== Definitions ==
A Diophantine equation is defined by a multi-variate polynomial <code>P(x,y,z,...)</code>. Solutions are values x,y,z,... such that <code>P(x,y,z,...) = 0</code>. The "magnitude" of a solution is max(|x|, |y|, |z|, ...) (the maximum absolute value of all it's variable assignments). The "score" for a Diophantine equation is the minimum magnitude across all solutions. In other words, it is the minimum N such that there exists a solution with -N ≤ x,y,z,... ≤ N.

The "size" or "height" of a Diophantine equation is the sum of <code>|coefficient|*2^degree</code> among all of its terms. For example <math>x^2 + xy + 6y + 1 = 0</math> has size <math>1 \cdot 2^2 + 1 \cdot 2^2 + 6 \cdot 2^1 + 1 \cdot 2^0 = 21 </math>.

<code>BBdio(H)</code> is the maximum score among all solvable polynomial Diophantine equations of height <code>H</code>.

== Champions ==
For 2 ≤ n ≤ 15: <code>BBdio(n) = n-2</code> via the trivial equation <math>x - (n-2) = 0</math>.[https://discord.com/channels/960643023006490684/1461768250575687897/1461769847754064036]
{| class="wikitable"
|+
!n
!BBdio(n)
!Champion
!Min Solution
!Source
|-
|2 ≤ n ≤ 20
|≥ n-2
|<math>x - (n-2) = 0</math>
|(n-2)
|
|-
|21
|≥ 26
|<math>x^2 + xy + 6y + 1 = 0</math>
|(-5, -26)
|
|-
|22
|≥ 22
|<math>x^3 + y^2 + 4 = 3x</math>
|(-8, ±22)
|
|-
|23
|≥ 31
|<math>x^2 + xy + x + 1 = 6y</math>
|(5, -31)
|
|-
|24
|≥ 66
|<math>x^3 + xy + 5y = 2</math>
|(-4, -66)
|
|-
|25
|≥ 470
|<math>x^3 + xy + y^2 + y + 7 = 0</math>
|(-63, -470)
|
|-
|26
|≥ 40
|<math>x^2 + xy + 7y + 4 = 0</math>
|
|
|-
|27
|≥ 849
|<math>x^3 + x^2 + y^2 + 9 = x</math>
|
|
|-
|28
|≥ 74
|<math>x^3 + xy + 2x + 5y = 2</math>
|
|
|-
|29
|> 10<sup>6</sup>
|<math>x^3 + x y^2 + x + z^3 - z + 1 = 0</math>
|(-4280795, 4360815, 5427173)*
|Andrew R. Booker via https://arxiv.org/abs/2108.08705 v3 page 23
|-
|...
|
|
|
|
|-
|48
|> 10<sup>10</sup>
|<math>x^3 + y^3 + z^3 = 24</math>
|(-2901096694, -15550555555, 15584139827)*
|https://oeis.org/A060467
|-
|57
|> 10<sup>15</sup>
|<math>x^3 + y^3 + z^3 = 33</math>
|(-2736111468807040, -8778405442862239, 8866128975287528)*
|https://oeis.org/A060467
|-
|66
|> 10<sup>16</sup>
|<math>x^3 + y^3 + z^3 = 42</math>
|(12602123297335631, 80435758145817515, −80538738812075974)*
|https://oeis.org/A060467
|}
Note: For large champions (marked with * in Min Solution column) these are not known to be strictly the smallest magnitude solutions, instead they appear to be solutions with the minimal smallest value. If that is the case, then this still bounds BBdio since the score of these equations must be ≥ the minimal smallest variable value.

== Cryptids ==
There are several open problems in the [[wikipedia:Sums_of_three_cubes|sums of three cubes]]. Specifically, it is not currently known if there are any integer solutions to the equations <math>x^3 + y^3 + z^3 = k</math> for k = 114, 390, 627, 633, 732, 921, or 975. Therefore, <math>x^3 + y^3 + z^3 = 114</math> is sort of like a BBdio(138) [[Cryptid]] in the sense that it requires solving an open math problem. However, this problem is expected to be solvable with perhaps 10x the compute used to solve the k=42, so it is not as futile as most Cryptids. Sheep suggests calling it an energy vampire.

== See Also ==

* [https://discord.com/channels/960643023006490684/1461768250575687897 Discord BBdio thread]

[[Category:Functions]]

Talk:Diophantine Equation

2026-08-14T07:39:11Z

C7X: /* Two relevant links */

==Two relevant links==
* "[https://mathoverflow.net/questions/400714/ Can you solve the listed smallest open Diophantine equations?]" may be a good source of cryptids, and it uses the same notion of size.
* "[https://web.archive.org/web/20130113072354/http://www.maths.bris.ac.uk/~maaib/baby/baby_project.pdf Studying the Atlas of all possible polynomial equations with quantifier-prefixes]" by Bovykin and De Smet, where they look for short (quantifier-prefixed) diophantine equation which gives behavior independent of some first-order theories. Also see the associated [https://googology.miraheze.org/wiki/Atlas_of_polynomials Googology Wiki article]. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 07:38, 14 August 2026 (UTC)

Talk:Diophantine Equation

2026-08-14T07:38:03Z

C7X: Created page with "==Two relevant links== * "[https://mathoverflow.net/questions/400714/ Can you solve the listed smallest open Diophantine equations?]" may be a good source of cryptids, and it uses the same notion of size. * "[https://web.archive.org/web/20130113072354/http://www.maths.bris.ac.uk/~maaib/baby/baby_project.pdf Studying the Atlas of all possible polynomial equations with quantifier-prefixes]" by Bovykin and De Smet, where they look for short (quantifier-prefixed) diophantine..."

==Two relevant links==
* "[https://mathoverflow.net/questions/400714/ Can you solve the listed smallest open Diophantine equations?]" may be a good source of cryptids, and it uses the same notion of size.
* "[https://web.archive.org/web/20130113072354/http://www.maths.bris.ac.uk/~maaib/baby/baby_project.pdf Studying the Atlas of all possible polynomial equations with quantifier-prefixes]" by Bovykin and De Smet, where they look for short (quantifier-prefixed) diophantine equation which gives behavior independent of some first-order theories. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 07:38, 14 August 2026 (UTC)

BB(2,5)

2026-08-05T21:22:06Z

C7X: Link abbreviation

The 2-state, 5-symbol Busy Beaver problem, '''BB(2,5)''', is unsolved. With the discovery of the [[Cryptids|Cryptid]] machine [[Hydra]] in April 2024, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(2,5) and thus [https://www.sligocki.com/2024/05/10/bb-2-5-is-hard.html BB(2,5) is Hard].

The current BB(2,5) [[Champions#5-Symbol TMs|champion]] {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}} was discovered by Daniel Yuan in June 2024, proving the lower bounds:

<math display="block">S(2,5) > \Sigma(2,5) > 10^{10^{10^{3\,314\,360}}} > 10 \uparrow\uparrow 4</math>

== Cryptids ==
Known Cryptids:

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA}}, known as [[Hydra]]
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}, known as the [[Bonus Cryptid]]
Potential Cryptids:

* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}. [https://discord.com/channels/960643023006490684/1354037062830919690/1354037062830919690 Shift overflow counter]
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}.
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}.
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}}. [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]
* {{TM|1RB2RA3LB---2LB_2LA0LA4RB0RB1LA}}. Probviously halting. 1/8 chance of beating champ.

==Top Halters==
The 20 longest running known halting BB(2,5) TMs are:
{| class="wikitable"
|+
!Standard format
!(approximate) runtime
!Discoverer
|-
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|<math>10 \uparrow\uparrow 4.8142742</math>
|Daniel Yuan
|-
|{{TM|1RB2LB4LB3LA1RZ_1LA3RA3LB0LB0RA|halt}}
|<math>>10^{38\,033}</math>
|Pavel Kropitz
|-
|{{TM|1RB2LA1RA2LB2LA_0LA2RB3RB4RA1RZ|halt}}
|<math>>1.9 \times 10^{704}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2RA3LA4RB---_2LA3RB3RA1LB3LB|halt}}
|<math>>8.3 \times 10^{466}</math> (lower bound given by score)
|Daniel Yuan
|-
|{{TM|1RB2LA4RA2LB2LA_0LA2RB3RB1RA1RZ|halt}}
|<math>>1.6 \times 10^{211}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2LA4RA2LB2LA_0LA2RB3RB4RA1RZ|halt}}
|<math>>1.6 \times 10^{211}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2LA4RA1LB2LA_0LA2RB3RB2RA1RZ|halt}}
|<math>>5.2 \times 10^{61}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB0RB4RA2LB2LA_2LA1LB3RB4RA1RZ|halt}}
|<math>>7 \times 10^{21}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}}
|<math>>6.04 \times 10^{21}</math>
|prurq and LegionMammal978
|-
|{{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}}
|<math>> 1.589 \times 10^{20}</math>
|prurq and LegionMammal978
|-
|{{TM|1RB1RZ4LA4LB2RA_2LB2RB3RB2RA0RB|halt}}
|<math>>9 \times 10^{16}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB3LA1LA0LB1RA_2LA4LB4LA1RA1RZ|halt}}
|<math>>3.77 \times 10^{16}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2RA1LA3LA2RA_2LA3RB4LA1LB1RZ|halt}}
|<math>>9 \times 10^{15}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2LB4RB2RB---_1LA3RA4RA3LB1LB|halt}}
|<math>6.64187 \times 10^{15}</math>
|mxdys
|-
|{{TM|1RB2RB4LB2LB---_2LA0LA3RB0LA2RA|halt}}
|<math>3.1317 \times 10^{15}</math>
|mxdys
|-
|{{TM|1RB0RA3LB1LB---_2LA3RB4RB3RA0LA|halt}}
|<math>2.7527 \times 10^{15}</math>
|mxdys
|-
|{{TM|1RB2LB2RA4LB1RA_1LA3RA3LA4LA---|halt}}
|<math>1.7828 \times 10^{15}</math>
|mxdys
|-
|{{TM|1RB3LA---4RB0LB_2LA3LB4LA1RB3RA|halt}}
|<math>6.66046 \times 10^{14}</math>
|mxdys
|-
|{{TM|1RB2RA1LA1LB3LB_2LA3RB1RZ4RA1LA|halt}}
|417,310,842,648,366
|Terry and Shawn Ligocki
|-
|{{TM|1RB3LA1LA2LA1RA_2LB3RA2RB4RB---|halt}}
|<math>1.61577 \times 10^{14}</math>
|mxdys
|}

== Certified progress ==
In April 2024, Shawn Ligocki publicly released a list of 23,411 undecided BB(2,5) machines. Justin Blanchard then made substantial progress over the course of the next month, reducing the list to 499 [[Holdouts lists|holdouts]] by late May 2024. In June 2024, @mxdys cut down the list to 273 using halting and inductive deciders, and again to 217 using [[Closed Tape Language|CTL]]. In February 2025, @mxdys ran a decider pipeline in Rocq that resulted in only 173 holdouts. Since then, additional machines have been proven in Rocq using both deciders and individual proofs.

On [https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 29 Mar 2025], @mxdys published a list of 83 holdouts that withstood state-of-the-art Rocq deciders.

Over the course of 5 months, @mxdys added 8 machines to Rocq<sup>[https://discord.com/channels/960643023006490684/1259770421046411285/1355799763437752521 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1355828077023854752 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1379521528869421137 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 <nowiki>[4]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 <nowiki>[5]</nowiki>]</sup>, lowering the certified holdout count to 75. There are 11 informal arguments, lowering the informal holdout count to 64.

Then, on [https://discord.com/channels/960643023006490684/1259770421046411285/1466208979511414885 29 Jan 2026], Andrew Ducharme found a machine nonhalting. This was verified by @mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1466331107279769736 the same day]. Hence the certified holdout count is 74, and there are still 11 informal arguments, with the informal holdout count being 63.

Later, on [https://discord.com/channels/960643023006490684/1259770421046411285/1471227102844944510 11 Feb 2026], Andrew Ducharme found another machine nonhalting, again verified by @mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471228798505582602 the same day]. @mxdys also [https://discord.com/channels/960643023006490684/1259770421046411285/1471229409829847111 announced another TM as a translated cycler], thus reducing the holdout count to 72, with 61 informal holdouts.

On [https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 11 March 2026], Peacemaker II solved a permutation of one of the TMs Andrew solved by tweaking some of the decider parameters. This result was verified by @mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1481326301209165877 the same day,] reducing the holdout count to 71, with 60 informal holdouts.

On 16 March 2026, mxdys formalised a [https://discord.com/channels/960643023006490684/1259770421046411285/1290449717536489622 proof from October 2024] and a [https://discord.com/channels/960643023006490684/1259770421046411285/1428501877947109437 proof from October 2025], thus reducing the holdout count to 69, with 60 informal holdouts.<sup>[https://discord.com/channels/960643023006490684/1259770421046411285/1483043448855461989 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1483043657778069564 <nowiki>[7]</nowiki>]</sup>

On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the holdout count to 67, with 60 informal holdouts.

On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys formalised] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.

On 30 May 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1510322250769760376 Andrew Ducharme showed] {{TM|1RB2LB---4LB0RB_1LA3RB4RB4RA1LB}} nonhalting via FAR. This machine previously had a nonhalting argument by Racheline, and [https://discord.com/channels/960643023006490684/1259770421046411285/1510867717194911874 mxdys verified the FAR solution in Rocq on 1 June], lowering the formal holdout count to 65.

== Holdouts ==
This section is based on the list of 83 holdouts published by @mxdys, and includes further progress as of 1 June 2026. There are 65 Rocq-holdouts, or 60 when considering informal proofs.

=== Cryptids ===

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}. Hydra
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}. Bonus Cryptid

=== Unsolved ===

* {{TM|1RB2RA3LA4LA2RB_2LA3RA---0RA1LA|undecided}}. Chaotic via [[Longitudinal Analysis|long. analysis]] - [https://discord.com/channels/960643023006490684/1259770421046411285/1436149296004071615 Notes by mxdys]
* {{TM|1RB2RA3LA4LA2RB_2LA3RB---0RA1LA|undecided}}. Chaotic via long. analysis
* {{TM|1RB---3RA2LA2RB_2LB3LA4LB4RA0RA|undecided}}. [https://discord.com/channels/960643023006490684/1471178503235043493/1471206925096980664 Does not halt in 1.25e13]
* {{TM|1RB3RB1LA2LA3RA_1LB2RA4RB0LA---|undecided}}. [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 Does not halt in 2e13 steps.]
* {{TM|1RB2RA4LA1RB4RB_1LB2LA3RA---0LB|undecided}}. [https://discord.com/channels/960643023006490684/1471178503235043493/1471206925096980664 Does not halt in 5e13 steps.]
* {{TM|1RB---4LB0LA4RA_2LB2LA3RA4LB0RB|undecided}}.
* {{TM|1RB4RA1LA4RB2LA_2LB3LA1RB2RA---|undecided}}.
* {{TM|1RB2RB3LA4LA1LA_2LB3RA---4RA1RB|undecided}}.
* {{TM|1RB3RB3LA4LA2RB_2LB3RA---1RA1LA|undecided}}.
* {{TM|1RB---3LB4RB0LA_2LB3LA3RB4RA0RA|undecided}}.
* {{TM|1RB3RB---4RA2RA_2LA2RA3LB4LB1LB|undecided}}. [https://discord.com/channels/960643023006490684/1492021157073916025/1492021157073916025 Longitudinal Analysis by dyuan.]
* {{TM|1RB3LA1LA2RB2RA_2LA4RA3LB1RA---|undecided}}. [https://discord.com/channels/960643023006490684/1491878887267762363/1491878887267762363 Longitudinal Analysis by dyuan.]
* {{TM|1RB4RB4RA1LA3LA_1LB2LA3RB2RB---|undecided}}. [https://discord.com/channels/960643023006490684/1489389585761828924/1489389585761828924 Longitudinal Analysis by dyuan.]
* {{TM|1RB2LA0RB4LB0LA_1LA3LA1RA4RA---|undecided}}. [https://discord.com/channels/960643023006490684/1471178503235043493/1471206925096980664 Does not halt in 1e13 steps.] [https://discord.com/channels/960643023006490684/1489363702560849920/1489363702560849920 Longitudinal Analysis by dyuan.]
* {{TM|1RB2LA0RB1LA3LB_1LA3LB1RA4RA---|undecided}}. Shift overflow mixed-digits counter - [https://discord.com/channels/960643023006490684/1440877223744770259/1440877223744770259 Analysis by hipparcos]
* {{TM|1RB2LA0RB4LB1RA_1LA3RA1RA---0LA|undecided}}. Shift overflow mixed-digits counter - [https://discord.com/channels/960643023006490684/1436181033992327333/1436181033992327333 Analysis by hipparcos] + [https://discord.com/channels/960643023006490684/1259770421046411285/1436151075450130443 mxdys's notes]
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}}. Potential Cryptid - [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}. Potential Cryptid
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}. Potential Cryptid
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}. Potential Cryptid - [https://discord.com/channels/960643023006490684/1354037062830919690/1354037062830919690 Shift overflow counter]
* {{TM|1RB2RA3LB---2LB_2LA0LA4RB0RB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1329808777754706046 30% chance of beating current champion]
* {{TM|1RB3LA1LA2RB2LB_1LB2RA4RA0RB---|undecided}}. [https://discord.com/channels/960643023006490684/1395820706050080869/1395820706050080869 Block analysis by @dyuan by "impurity score"]
* {{TM|1RB---4LB1RA4RA_2LB2LA3RA4LB0RB|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1378248683161653289 Analysis by Andrew Ducharme and @mxdys]
* {{TM|1RB---4RB2RB4LA_2LB3LA3LB0RA0RB|undecided}}. [https://discord.com/channels/960643023006490684/1353983911222312970/1353983911222312970 Bouncer + chaotic counter]
* {{TM|1RB2LA0RB0LB3LB_2LA4RB3RA0RA---|undecided}}. [https://discord.com/channels/960643023006490684/1395872756268269668/1395872756268269668 Analysis by Peacemaker II]
* {{TM|1RB2RA3LB4LA---_2LA0RB1LA2RB0RA|undecided}}. [https://discord.com/channels/960643023006490684/1348878717870673981 Analysis by @dyuan01 and @Legion]
* {{TM|1RB2RA3LA---2LB_2LA4RA4RB0RB0LA|undecided}}. Spaghetti, [https://discord.com/channels/960643023006490684/1344221797020602398/1344221797020602398 analysis by @nerdyjoe], [https://discord.com/channels/960643023006490684/1471178503235043493/1471206925096980664 does not halt in 4e13.]
* {{TM|1RB3LA3LB0RB0LA_2LA4RB1LB1RA---|undecided}}. Permutation of "Spaghetti TM", [https://discord.com/channels/960643023006490684/1344221797020602398 analysis by nerdyjoe]
* {{TM|1RB2RA3LA4LA2RB_2LA---3LB1RA3RA|undecided}}. [https://discord.com/channels/960643023006490684/1353983911222312970/1355112650690003028 Bouncer + chaotic counter]
* {{TM|1RB3LA3LA0RB2LB_2LA4LA4RA2RA---|undecided}}. [https://discord.com/channels/960643023006490684/1376383949575557161 Analysis by @mxdys]
* {{TM|1RB2LB3LA0RA1LB_2LA4RA3RB3LA---|undecided}}. [https://discord.com/channels/960643023006490684/1344221797020602398/1344221797020602398 Analysis by @nerdyjoe]
* {{TM|1RB3LB0RB---2LB_2LA3RA4RB2RB0LA|undecided}}. [https://discord.com/channels/960643023006490684/1376577295938097253 Analysis by @mxdys]
* {{TM|1RB3LB4LA0LB---_2LA0LA1RB0RA3RA|undecided}}. [https://discord.com/channels/960643023006490684/1395872756268269668/1395872756268269668 Analysis by Peacemaker II]
* {{TM|1RB3RB1LB2RA---_2LA2RB1LA4LB0RA|undecided}}. [https://discord.com/channels/960643023006490684/1397318518961082398 Analysis by Legion and @dyuan]
* {{TM|1RB2LA0RB---4LA_1LA3LA1RA4RA1LB|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1329629220795715674 Analysis by @racheline]
* {{TM|1RB4LA1RA1RB1LA_2LB3LA---4RA2RB|undecided}}. [https://discord.com/channels/960643023006490684/1359561443929886760 Basic long. analysis by @dyuan]
* {{TM|1RB3RA3RB4LA1LA_1LB2LA1LA---1RB|undecided}}. [https://discord.com/channels/960643023006490684/1349602897024782358 Long. analysis by @dyuan suggests chaotic, potentially halt]
* {{TM|1RB2LA4LA1RA1LA_2LB3RB4RB---2RA|undecided}}. [https://discord.com/channels/960643023006490684/1084047886494470185/1255570421437169805 Long. analysis rules by @Legion, ran to cell 155 without halting]
* {{TM|1RB2RB4LA2RA1LA_2LA4RA3LA---3RA|undecided}}. Chaotic via long. analysis. [https://discord.com/channels/960643023006490684/1353983911222312970/1353987502062702622 Probviously nonhalting]
* {{TM|1RB2LA4RA1LA3LA_0LA2RB3RB2LB---|undecided}}. 1D CA-like. [https://discord.com/channels/960643023006490684/1354107790330691655 Analysis by @dyuan and @mxdys]
* {{TM|1RB2LA4RA1LA3LA_0LA3RB3LB2RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA1LA4RA2LA_0LA3RB3LB2RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA3LA4RA1LA_0LA3LB3RB1RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA3LB4LB---_0LA4LB3RA4LA0RB|undecided}}. Fractal?
14 grandchildren of {{TM|1RB2LA0RB1LB_1LA3RA1RA---|undecided}}

* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4RB0LB|undecided}}.

and the family 1RB2LA0RB1LB---_1LA3RA1RA4LB---. See [https://discord.com/channels/960643023006490684/1336734852308799579 this thread] for more details.

* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2RB|undecided}}. Simulated for <math>~9*10^{1167}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2LB|undecided}}. Simulated for <math>~1.3*10^{1094}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1RB|undecided}}. Simulated for <math>~9.8*10^{1226}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1LB|undecided}}. Simulated for <math>~3*10^{1140}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0LB|undecided}}. Simulated for <math>~2.6*10^{889}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0RB|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB3RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2LA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1LA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0LA|undecided}}.

=== Solved with moderate rigor ===

* {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|undecided}}. [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Longitudinal analysis by @Legion implies halting], [https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 simulated until halt by prurq using Quick_Sim]
* {{TM|1RB3LA4LA1LA2RA_2LA4RB---0RA0LA|undecided}}. [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Longitudinal analysis by @Legion implies halting]
* {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}}. [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Longitudinal analysis by @Legion implies halting], [https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 simulated until halt by prurq using Quick_Sim]
* {{TM|1RB3RA2LB1LB1RB_2LA2RA4LA1LA---|undecided}}. [[Dekaheptoid]] - [https://discord.com/channels/960643023006490684/1259770421046411285/1267650177389432913 Unverified nonhalting proof by @dyuan]
* {{TM|1RB3RB1LB---2RB_2LA1RA4LB2LA2RA|undecided}}. [[Dekaheptoid]] - [https://discord.com/channels/960643023006490684/1259770421046411285/1267650177389432913 Unverified nonhalting proof by @dyuan]

=== Formally proven ===

* {{TM|1RB2RA3LA4LA2RB_2LA0RA---0RA1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1355828077023854752 Rocq-decided by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1355714495326060706 Longitudinal analysis by Legion implies nonhalting]
* {{TM|1RB2RA3LA4RB---_2LA3RB3RA1LB3LB|halt}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 Rocq-decided by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1373347187836194898 Halting argument by @dyuan]
* {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 Rocq-decided by @mxdys]. Current champion
* {{TM|1RB3RA4LB2RA2RB_2LA---3LA0LB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 Rocq-decided by @mxdys.]
* {{TM|1RB3RB---0RA2RB_2LA4RA3LB1LB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 Rocq-decided by @mxdys.]
* {{TM|1RB0LB2LA4LB3LA_2LA---3RA4RB2RB|undecided}}. [https://discord.com/channels/960643023006490684/1375251026411786310/1375556785603084469 Rocq-decided by @mxdys]
* {{TM|1RB2LA0LB1LA2RA_0LA3RA1RA4LB---|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1428501877947109437 Nonhalting argument by Peacemaker II]. [https://discord.com/channels/960643023006490684/1259770421046411285/1483043657778069564 Confirmed in Rocq by @mxdys].
* {{TM|1RB2LA1RA---1LA_1LA4RB3LB0RB2RB|undecided}}. [https://discord.com/channels/960643023006490684/1375251026411786310/1375556785603084469 Rocq-decided by @mxdys]
* {{TM|1RB2LA3LA4RA0LA_1LA3RB1RB1LB---|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379521528869421137 Rocq-decided by @mxdys]
* {{TM|1RB2LA0RB1LB0LB_1LA3RA1RA4RA---|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1466208979511414885 Non-halting found by Andrew Ducharme]. [https://discord.com/channels/960643023006490684/1259770421046411285/1466331107279769736 Confirmed in Rocq by @mxdys]. Grandchild of {{TM|1RB2LA0RB1LB_1LA3RA1RA---|undecided}}.
* {{TM|1RB2RB---0LB3LA_2LA2LB3RB4RB1LB|undecided}}. Chaotic via long. analysis. [https://discord.com/channels/960643023006490684/1378560417235734558 Analysis of permutation by mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1471227102844944510 Non-halting found by Andrew Ducharme]. [https://discord.com/channels/960643023006490684/1259770421046411285/1471228798505582602 Confirmed in Rocq by @mxdys.]
* {{TM|1RB3LA1RA4LA2RA_2LA---1LA0RA3RB|undecided}}. Chaotic via long. analysis. [https://discord.com/channels/960643023006490684/1378560417235734558 More analysis by mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1472647706835943596 High-level behaviour by Peacemaker II.] [https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 Non-halting found by Peacemaker II.] [https://discord.com/channels/960643023006490684/1259770421046411285/1481326301209165877 Confirmed in Rocq by @mxdys.]
* {{TM|1RB3LA1LA4LA2RA_2LB2RA---0RA0RB|undecided}}. - [https://discord.com/channels/960643023006490684/1259770421046411285/1436151969986379868 Notes by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1471229409829847111 Translated cycler via @mxdys.]
* {{TM|1RB4LA1LB2LA0RB_2LB3RB4LA---1RA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1290449717536489622 Nonhalting argument by @dyuan]. [https://discord.com/channels/960643023006490684/1259770421046411285/1483043448855461989 Confirmed in Rocq by @mxdys].
* {{TM|1RB1RB3LA4LA2RA_2LB3RA---3RA4RB|undecided}}. [[Beaver Math Olympiad#3. 1RB0RB3LA4LA2RA 2LB3RA---3RA4RB (bbch) and 1RB1RB3LA4LA2RA 2LB3RA---3RA4RB (bbch)|BMO problem 3]] by @dyuan. [https://discord.com/channels/960643023006490684/1259770421046411285/1488743526882738276 Confirmed in Lean by @mammillaria] using [https://aristotle.harmonic.fun/ Aristotle AI], [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 translated to Rocq by @mxdys] using other LLM.
* {{TM|1RB0RB3LA4LA2RA_2LB3RA---3RA4RB|undecided}}. BMO problem 3 by @dyuan. [https://discord.com/channels/960643023006490684/1259770421046411285/1488743526882738276 Confirmed in Lean by @mammillaria] using [https://aristotle.harmonic.fun/ Aristotle AI], [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 translated to Rocq by @mxdys] using other LLM.
* {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB|undecided}}. BMO problem 3 variant - [https://discord.com/channels/960643023006490684/1259770421046411285/1415337575543214132 Nonhalting argument by @dyuan], [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 translated to Rocq by @mxdys] using an LLM.
* {{TM|1RB2LB---4LB0RB_1LA3RB4RB4RA1LB|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1329663999700111471 Nonhalting argument by @racheline]. [https://discord.com/channels/960643023006490684/1259770421046411285/1510322250769760376 FAR solution by Andrew Ducharme], [https://discord.com/channels/960643023006490684/1259770421046411285/1510867717194911874 Rocq by mxdys.]

[[Category:BB Domains]][[Category:BB(2,5)]]

Holdouts lists

2026-08-05T21:20:03Z

C7X: Add one more /* BB(2,5) */

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
|1064
|15,743,264
|-
!3-symbol
|0
|4
|4,747,566
|
|
|
|-
!4-symbol
|0
|11,362,197
|
|
|
|
|-
!5-symbol
|65
|
|
|
|
|
|-
!6-symbol
|412,086
|
|
|
|
|
|}

== 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.

=== BB(5) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|July 2nd, 2024
|bbchallenge
|0
|[https://cable.ayra.ch/empty/?id=0 List of 0 holdouts]
|'''BB(5) is solved!'''
|-
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

=== BB(6) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1531681818439319604 July 28th, 2026]
|@mxdys
|1064
|[https://wiki.bbchallenge.org/w/images/e/e9/BB6_holdouts_1064.txt BB6_holdouts_1064.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1521203430046171146 June 29th, 2026]
|@mxdys
|1094
|[https://wiki.bbchallenge.org/w/images/9/90/BB6_holdouts_1094.txt BB6_holdouts_1094.txt]
|
|-
|[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
|-
|[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
|-
|[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]
|
|-
|[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]
|
|-
|[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]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 January 29th, 2026]
|@mxdys
|1314
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 BB6_holdouts_1314.txt]
|
|-
|[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]
|
|-
|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.
|-
|[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]
|
|-
|[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]
|
|-
|[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]
|
|-
|[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]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 September 21, 2025]
|@mxdys
|1691
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 BB6_holdouts_1691.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 July 13, 2025]
|@mxdys
|2891
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 BB6_holdouts_2891.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 July 5, 2025]
|@mxdys
|3094
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 BB6_holdouts_3094.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 June 18, 2025]
|@mxdys
|3335
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 BB6_holdouts_3335.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 March 1, 2025]
|@mxdys
|3571
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 BB6_holdouts_3571.txt]
|
|-
|[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.
|-
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 October 19, 2024]
|@mxdys
|4521
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 BB6_holdouts_4521.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 September 23, 2024]
|@mxdys
|4741
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 BB6_holdouts_4741.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 September 7, 2024]
|@mxdys
|4986
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 BB6_holdouts_4986.txt]
|
|-
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 July 28, 2024]
|@mxdys
|6417
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 BB6_holdouts_6417.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 July 15, 2024]
|@mxdys
|7013
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 BB6_holdouts_7013.txt]
|
|-
|[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
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[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
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 July 5, 2024]
|@mxdys
|7449
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 BB6_holdouts_7449.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 June 28, 2024]
|@mxdys
|9366
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 BB6_holdouts_9366.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 June 24, 2024]
|@Justin Blanchard
|9853
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 BB6_holdouts_9853.txt]
|
|-
|[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
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 June 20, 2024]
|@mxdys
|10,944
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 BB6_holdouts_10944.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
|12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 May 28, 2024]
|@mxdys
|18,560
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 BB6_holdouts_18560.txt]
|
|-
|[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>
|-
|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)
|-
|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(7) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://drive.google.com/drive/folders/1_lIqfvj4_J7WWl5LOBUp_pntoI99QYui May 25th, 2026]
|
|17,823,260
|[https://drive.google.com/file/d/1GFumm3tzm58Yxza_sD_BdJ3RMY_DKmRv/view 7x2_holdouts_17823260.txt.zip]
|Work done by Andrew Ducharme
|-
|[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]
|
|-
|[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(3,3) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|[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
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|}

=== BB(4,3) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- May 9th, 2026]
|@Andrew Ducharme
|5,127,263
|[https://drive.google.com/file/d/1xHukBgJnm1zctZkOMR0UzdQyzr4Fx5pC/view 4x3_holdouts_5127263.txt.zip]
|Work done by Andrew Ducharme
|-
|[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
|-
|[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]
|
|-
|[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]
|
|-
|[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(3,4) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://drive.google.com/drive/folders/1uGWJMYYuVGJj581GX_vlm3Wv1adYbcRs May 11th, 2026]
|@XnoobSpeakable
|11,362,197
|[https://drive.google.com/file/d/1-Fnexjfq0mWyZDPa2MNFmU64EpSu-3Ig/view 3x4_holdouts_11362197.zip]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]]
|-
|[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
|-
|[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]].
|-
|[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]].
|-
|[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(2,5) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29, 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 April 25, 2024]
|@Justin Blanchard
|920
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 April 24, 2024]
|@Justin Blanchard
|1045
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 April 22, 2024]
|@Justin Blanchard
|1588
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 April 19, 2024]
|@Justin Blanchard
|1835
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 April 18, 2024]
|@Justin Blanchard
|1921
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 April 17, 2024]
|@Justin Blanchard
|2354
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 April 16, 2024]
|@Justin Blanchard
|2438
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 April 15, 2024]
|@Justin Blanchard
|2564
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 April 12, 2024]
|@Justin Blanchard
|3364
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 2x5.todo.txt]
|
|}

=== BB(2,6) ===
{| class="wikitable sortable"
|+
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[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.
|-
|[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
|-
|[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
|-
|[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]
|}

== 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(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout August 5, 2026]
|@-d
|13
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_13_unofficial.txt sz23_13_unofficial.txt]
|Unofficial holdouts list
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout July 25, 2026]
|@-d
|694
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_694_unofficial.txt sz23_694_unofficial.txt]
|Unofficial holdouts list, reduction by Claude
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout July 25, 2026]
|@-d
|703
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_703_unofficial.txt sz23_703_unofficial.txt]
|Unofficial holdouts list, reduction by Claude
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout July 24, 2026]
|@-d
|8,021
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_8021_unofficial.txt sz23_8021_unofficial.txt]
|Unofficial holdouts list, reduction by Claude
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout July 11, 2026]
|@-d
|21233
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_21233.txt sz23_21233.txt]
|
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 3, 2026]
|@-d
|21295
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_21295.txt sz23_21295.txt]
|Programs simulated to halting were removed
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 2, 2026]
|@-d
|21320
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_21320.txt sz23_21320.txt]
|Programs simluated to halting were removed
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 2, 2026]
|@-d
|28100
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_28100.txt sz23_28100.txt]
|
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 2, 2026]
|@-d
|29188
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_29188.txt sz23_29188.txt]
|
|-
|BBf(23)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1510749997392724190 May 31, 2026]
|@-d
|29250
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_29250.txt sz23_29250.txt]
|
|-
|BBf(23)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1510749997392724190 May 31, 2026]
|@-d
|94367
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_94367.txt sz23_94367.txt]
|
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout May 30, 2026]
|@-d
|790,335
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_790335.txt sz23_790335.txt]
|
|-
|BBf(22)
|[https://github.com/int-y1/BBFractran/tree/main/holdout April 4, 2026]
|@-d
|3
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_3.txt sz22_3.txt]
|Holdout reduction by Claude Opus 4.6
|-
|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/960643023006490684/1243312334907375676/1533832727441178644 August 3, 2026]
|@creeperman7002
|630
|[https://discord.com/channels/960643023006490684/1243312334907375676/1533832727441178644 TT2x4_holdouts_630.txt]
|
|-
|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(42)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1517130491512492083 June 18, 2026]
|@C++DSucker
|Variable (240)
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1544675322#gid=1544675322 Spreadsheet]
|
|-
|BBλ(41)
|
|
|Variable (6)
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1502335946#gid=1502335946 Spreadsheet]
|
|-
|BBλ(40)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1514278632695005224 June 10, 2026]
|@C++DSucker
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=741510300#gid=741510300 Spreadsheet]
|'''BBλ(40) was solved on July 20th, 2026'''
|-
|BBλ(39)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1492187654429868062 April 10, 2026]
|@Sligocki
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1717120706#gid=1717120706 Spreadsheet]
|'''BBλ(39) was solved on June 10th, 2026'''
|-
|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]
|
|}

=== CounterScript ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBCS(11)
|[https://github.com/AzertyWasTaken/CounterScript/tree/main/Holdouts July 10, 2026]
|@Azerty
|1
|[https://github.com/AzertyWasTaken/CounterScript/blob/main/Holdouts/11.txt 11.txt]
|The holdout may be resolved by hand.
|}
== References ==

Holdouts lists

2026-08-05T18:18:53Z

C7X: End domain with length-0 list since it's solved

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
|1064
|15,743,264
|-
!3-symbol
|0
|4
|4,747,566
|
|
|
|-
!4-symbol
|0
|11,362,197
|
|
|
|
|-
!5-symbol
|65
|
|
|
|
|
|-
!6-symbol
|412,086
|
|
|
|
|
|}

== 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.

=== BB(5) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|-
|July 2nd, 2024
|bbchallenge
|0
|[https://cable.ayra.ch/empty/?id=0 List of 0 holdouts]
|'''BB(5) is solved!'''
|}

=== BB(6) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1531681818439319604 July 28th, 2026]
|@mxdys
|1064
|[https://wiki.bbchallenge.org/w/images/e/e9/BB6_holdouts_1064.txt BB6_holdouts_1064.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1521203430046171146 June 29th, 2026]
|@mxdys
|1094
|[https://wiki.bbchallenge.org/w/images/9/90/BB6_holdouts_1094.txt BB6_holdouts_1094.txt]
|
|-
|[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
|-
|[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
|-
|[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]
|
|-
|[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]
|
|-
|[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]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 January 29th, 2026]
|@mxdys
|1314
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 BB6_holdouts_1314.txt]
|
|-
|[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]
|
|-
|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.
|-
|[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]
|
|-
|[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]
|
|-
|[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]
|
|-
|[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]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 September 21, 2025]
|@mxdys
|1691
|[https://discord.com/channels/960643023006490684/1239205785913790465/1419290767767240734 BB6_holdouts_1691.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 July 13, 2025]
|@mxdys
|2891
|[https://discord.com/channels/960643023006490684/1239205785913790465/1393808719615361205 BB6_holdouts_2891.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 July 5, 2025]
|@mxdys
|3094
|[https://discord.com/channels/960643023006490684/1239205785913790465/1390953522379489401 BB6_holdouts_3094.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 June 18, 2025]
|@mxdys
|3335
|[https://discord.com/channels/960643023006490684/1239205785913790465/1384957116376158228 BB6_holdouts_3335.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 March 1, 2025]
|@mxdys
|3571
|[https://discord.com/channels/960643023006490684/1239205785913790465/1345246656769163284 BB6_holdouts_3571.txt]
|
|-
|[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.
|-
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 October 19, 2024]
|@mxdys
|4521
|[https://discord.com/channels/960643023006490684/1239205785913790465/1297163115179278336 BB6_holdouts_4521.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 September 23, 2024]
|@mxdys
|4741
|[https://discord.com/channels/960643023006490684/1239205785913790465/1287580515938865225 BB6_holdouts_4741.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 September 7, 2024]
|@mxdys
|4986
|[https://discord.com/channels/960643023006490684/1239205785913790465/1282002734912110655 BB6_holdouts_4986.txt]
|
|-
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 July 28, 2024]
|@mxdys
|6417
|[https://discord.com/channels/960643023006490684/1239205785913790465/1267013907910758451 BB6_holdouts_6417.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 July 15, 2024]
|@mxdys
|7013
|[https://discord.com/channels/960643023006490684/1239205785913790465/1262276904887521300 BB6_holdouts_7013.txt]
|
|-
|[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
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[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
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 July 5, 2024]
|@mxdys
|7449
|[https://discord.com/channels/960643023006490684/1239205785913790465/1258810921879605298 BB6_holdouts_7449.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 June 28, 2024]
|@mxdys
|9366
|[https://discord.com/channels/960643023006490684/1239205785913790465/1256254912573079572 BB6_holdouts_9366.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 June 24, 2024]
|@Justin Blanchard
|9853
|[https://discord.com/channels/960643023006490684/1239205785913790465/1254684723955896362 BB6_holdouts_9853.txt]
|
|-
|[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
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 June 20, 2024]
|@mxdys
|10,944
|[https://discord.com/channels/960643023006490684/1239205785913790465/1253466035189846056 BB6_holdouts_10944.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
|12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 May 28, 2024]
|@mxdys
|18,560
|[https://discord.com/channels/960643023006490684/1239205785913790465/1245116022873264129 BB6_holdouts_18560.txt]
|
|-
|[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>
|-
|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)
|-
|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(7) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://drive.google.com/drive/folders/1_lIqfvj4_J7WWl5LOBUp_pntoI99QYui May 25th, 2026]
|
|17,823,260
|[https://drive.google.com/file/d/1GFumm3tzm58Yxza_sD_BdJ3RMY_DKmRv/view 7x2_holdouts_17823260.txt.zip]
|Work done by Andrew Ducharme
|-
|[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]
|
|-
|[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(3,3) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|[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
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|}

=== BB(4,3) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW- May 9th, 2026]
|@Andrew Ducharme
|5,127,263
|[https://drive.google.com/file/d/1xHukBgJnm1zctZkOMR0UzdQyzr4Fx5pC/view 4x3_holdouts_5127263.txt.zip]
|Work done by Andrew Ducharme
|-
|[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
|-
|[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]
|
|-
|[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]
|
|-
|[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(3,4) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://drive.google.com/drive/folders/1uGWJMYYuVGJj581GX_vlm3Wv1adYbcRs May 11th, 2026]
|@XnoobSpeakable
|11,362,197
|[https://drive.google.com/file/d/1-Fnexjfq0mWyZDPa2MNFmU64EpSu-3Ig/view 3x4_holdouts_11362197.zip]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]]
|-
|[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
|-
|[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]].
|-
|[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]].
|-
|[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(2,5) ===
{| class="wikitable sortable"
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 April 25, 2024]
|@Justin Blanchard
|920
|[https://discord.com/channels/960643023006490684/1084047886494470185/1233154495635259543 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 April 24, 2024]
|@Justin Blanchard
|1045
|[https://discord.com/channels/960643023006490684/1084047886494470185/1232667326126489710 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 April 22, 2024]
|@Justin Blanchard
|1588
|[https://discord.com/channels/960643023006490684/1084047886494470185/1231941302975987844 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 April 19, 2024]
|@Justin Blanchard
|1835
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230737657076650066 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 April 18, 2024]
|@Justin Blanchard
|1921
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230491507287523409 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 April 17, 2024]
|@Justin Blanchard
|2354
|[https://discord.com/channels/960643023006490684/1084047886494470185/1230132897583857756 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 April 16, 2024]
|@Justin Blanchard
|2438
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229770471898087506 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 April 15, 2024]
|@Justin Blanchard
|2564
|[https://discord.com/channels/960643023006490684/1084047886494470185/1229406059274305546 2x5.todo.txt]
|
|-
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 April 12, 2024]
|@Justin Blanchard
|3364
|[https://discord.com/channels/960643023006490684/1084047886494470185/1228351493745152083 2x5.todo.txt]
|
|}

=== BB(2,6) ===
{| class="wikitable sortable"
|+
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[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.
|-
|[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
|-
|[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
|-
|[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]
|}

== 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(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout August 5, 2026]
|@-d
|13
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_13_unofficial.txt sz23_13_unofficial.txt]
|Unofficial holdouts list
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout July 25, 2026]
|@-d
|694
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_694_unofficial.txt sz23_694_unofficial.txt]
|Unofficial holdouts list, reduction by Claude
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout July 25, 2026]
|@-d
|703
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_703_unofficial.txt sz23_703_unofficial.txt]
|Unofficial holdouts list, reduction by Claude
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout July 24, 2026]
|@-d
|8,021
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_8021_unofficial.txt sz23_8021_unofficial.txt]
|Unofficial holdouts list, reduction by Claude
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout July 11, 2026]
|@-d
|21233
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_21233.txt sz23_21233.txt]
|
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 3, 2026]
|@-d
|21295
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_21295.txt sz23_21295.txt]
|Programs simulated to halting were removed
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 2, 2026]
|@-d
|21320
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_21320.txt sz23_21320.txt]
|Programs simluated to halting were removed
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 2, 2026]
|@-d
|28100
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_28100.txt sz23_28100.txt]
|
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout June 2, 2026]
|@-d
|29188
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_29188.txt sz23_29188.txt]
|
|-
|BBf(23)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1510749997392724190 May 31, 2026]
|@-d
|29250
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_29250.txt sz23_29250.txt]
|
|-
|BBf(23)
|[https://discord.com/channels/960643023006490684/1438019511155691521/1510749997392724190 May 31, 2026]
|@-d
|94367
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_94367.txt sz23_94367.txt]
|
|-
|BBf(23)
|[https://github.com/int-y1/BBFractran/tree/main/holdout May 30, 2026]
|@-d
|790,335
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz23_790335.txt sz23_790335.txt]
|
|-
|BBf(22)
|[https://github.com/int-y1/BBFractran/tree/main/holdout April 4, 2026]
|@-d
|3
|[https://github.com/int-y1/BBFractran/blob/main/holdout/sz22_3.txt sz22_3.txt]
|Holdout reduction by Claude Opus 4.6
|-
|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/960643023006490684/1243312334907375676/1533832727441178644 August 3, 2026]
|@creeperman7002
|630
|[https://discord.com/channels/960643023006490684/1243312334907375676/1533832727441178644 TT2x4_holdouts_630.txt]
|
|-
|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(42)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1517130491512492083 June 18, 2026]
|@C++DSucker
|Variable (240)
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1544675322#gid=1544675322 Spreadsheet]
|
|-
|BBλ(41)
|
|
|Variable (6)
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1502335946#gid=1502335946 Spreadsheet]
|
|-
|BBλ(40)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1514278632695005224 June 10, 2026]
|@C++DSucker
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=741510300#gid=741510300 Spreadsheet]
|'''BBλ(40) was solved on July 20th, 2026'''
|-
|BBλ(39)
|[https://discord.com/channels/960643023006490684/1355653587824283678/1492187654429868062 April 10, 2026]
|@Sligocki
|0
|[https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1717120706#gid=1717120706 Spreadsheet]
|'''BBλ(39) was solved on June 10th, 2026'''
|-
|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]
|
|}

=== CounterScript ===
{| class="wikitable sortable"
|+
!Domain
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|BBCS(11)
|[https://github.com/AzertyWasTaken/CounterScript/tree/main/Holdouts July 10, 2026]
|@Azerty
|1
|[https://github.com/AzertyWasTaken/CounterScript/blob/main/Holdouts/11.txt 11.txt]
|The holdout may be resolved by hand.
|}
== References ==

Oracle

2026-06-10T21:15:46Z

C7X: This page is under construction, but I haven't seen any sources myself with these oracles (is an oracle the same thing semantically as a subset of N?)

An oracle is a black box that can give an answer to any instance of a certain problem in a single operation. The problem ⁠can be of any complexity class, or it can even be an undecidable problem such as the halting problem.

* 0': decide the halting problem for computable programs.
* 1': decide the halting problem for programs containing 0'
* Ω': decide the halting problem of any program, including programs containing Ω'{{citation needed}}
* Ω2': decide if a programs is well-founded (do not create any kind of paradox){{citation needed}}

PAGE IN CONSTRUCTION

Graham's number

2026-05-22T05:49:28Z

C7X: Looks like Wythagoras originally had a 19-state machine, then <4 hours later improved it to 18 states /* History of Graham-beating TMs */

'''Graham's number''' (<math>g_{64}</math> or <math>G</math>) is a famously huge number which Martin Gardner claimed was the "largest number ever used in a serious mathematical proof" in 1977. Since it is one of the most famous large numbers, it has become a bit of a yardstick for measuring "hugeness". In the specific context of the Busy Beaver game, we can ask, what is the smallest <math>n</math> such that <math>BB(n) > g_{64}</math>. There is an active search for the smallest TM that runs for over Graham's number steps.

== Definition ==
See [https://en.wikipedia.org/wiki/Graham%27s_number Wikipedia article] for more detail
<math display="block">
\begin{array}{l}
g_0 & = & 4 \\
g_n & = & 3 \uparrow^{g_{n-1}} 3 & \text{if } n \ge 1 \\
\end{array}
</math>

Graham's number is <math>g_{64}</math>.

Using the [[fast-growing hierarchy]], <math>g_{64} < f_{\omega + 1}(64)</math>.

Let <math>N_G</math> be the smallest integer such that <math>S(N_G) > g_{64}</math>.

== Bounds ==
The current known bounds for <math>N_G</math> are:
<math display="block">
6 \le N_G \le 13
</math>

The lower bound comes from the proof that [[BB(5)]] = 47,176,870 and the upper bound from a specific 13-state TM found by 50_ft_lock in March 2026 which runs for <math> > f_{\omega + 1}(2\,046) > g_{64} </math> steps.

=== History of Graham-beating TMs ===
There is no one authoritative source on the history of TMs beating Graham's number. Most were posted in personal blog posts, Googology pages/comments or on the bbchallenge Discord. They are often unverified and sometimes undocumented. This list is based upon historical accounts listed on [https://googology.fandom.com/wiki/Graham%27s_number#Comparison_with_busy_beaver_function Googologogy wiki], [https://cs.stackexchange.com/questions/69469/what-is-the-smallest-n-such-that-bbn-grahams-number/69476#69476 a 2017 CS Stack Exchange answer], [https://www.sligocki.com/2010/07/04/beating-grahams-number.html#results-from-the-future a 2022 history synthesis by Shawn Ligocki] and [https://bbchallenge.org/~pascal.michel/ha#topdown summary by Pascal Michel]. In general, these results are self-reported and we do not know of independent verification for most of them. Most of these were actually proven as bounds for the <math>\Sigma</math> function, but since <math>S(n) \ge \Sigma(n)</math> they apply to this formulation as well.
{| class="wikitable"
|+History of Graham-beating TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|91
|9 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|72
|13 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|64
|19 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/64-state-turing-machine-whose-output.html Blog Post] [https://web.archive.org/web/20130509222047/https://sites.google.com/site/res0001/surpassing-graham-s-number Summary]
|
|-
|25
|31 Mar 2013
|"Deedlit11"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines#A_New_Record!_Beating_Graham's_number_with_a_2-symbol_TM Googology Post]
|
|-
|24
|27 September 2013
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines?commentId=4400000000000011880 Googology Comment]
|
|-
|23
|7 October 2013
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines?commentId=4400000000000011880&replyId=4400000000000036301 Googology Comment]
|
|-
|22
|5 August 2014
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/NEWS!_I_found_a_22-state_machine_that_beats_G! Googology Post]
|
|-
|19
|24 Jul 2016
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham's_Number!?useskin=oasis Googology Post]
|
|-
|18
|24 Jul 2016
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham's_Number!?useskin=oasis Googology Post]
|
|-
|16
|26 Mar 2021
|Daniel Nagaj
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham%27s_Number!?commentId=4400000000000019187&replyId=4400000000000101283 Googology Comment] [http://morphett.info/turing/turing.html?197640ce0f380f8a6b0a4cdd138156a0 TM Definition]
|[https://www.sligocki.com/2022/07/11/bb-16-graham.html Analysis by Shawn Ligocki in 2022]
|-
|14
|17 Aug 2024
|[[User:Racheline|Racheline]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Discord Message]
|
|-
|13
|13 March 2026
|50_ft_lock
|[https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 Discord Message]
|}

BB(2,3)

2026-05-20T22:35:41Z

C7X: Add citation from Michel's page

'''BB(2,3)''' refers to the [[Busy Beaver function]] with 2 states and 3 symbols. The value was discovered by Brady in 1988 (and independently by Michel in 2004) and proven by Lafitte and Papazian in 2007.<ref>G. Lafitte, C. Papazian, "[http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.104.3021&rep=rep1&type=pdf#page=231 The Fabric of Small Turing Machines]". In ''Computation and Logic in the Real World, Proceedings of the Third Conference on Computability in Europe'', June 2007, pp. 219--227.</ref>

== Champions ==
S(2,3) = 38 and Σ(2,3) = 9 and both are achieved by only one [[Champions#3-Symbol TMs|champion]] (in [[TNF]]):
* {{TM|1RB2LB1RZ_2LA2RB1LB|halt}}

== Enumeration ==
The top 20 longest running BB(2,3) TMs (in [[TNF-1RB]]) are:
Standard Format Steps Σ
1RB2LB1RZ_2LA2RB1LB 38 9
1RB0LB1RZ_2LA1RB1RA 29 8
1RB2LA1RZ_1LB1LA0RA 26 6
1RB1LA1LB_0LA2RA1RZ 26 6
1RB1LB1RZ_2LA2RB1LB 24 7
1RB2LA1RZ_0LB1LA0RA 22 4
1RB2RB1RZ_1LB1RA0LA 21 4
1RB2LA2RB_1LA1RZ1RA 20 6
1RB2LA1RZ_2LA2RB1LB 20 6
1RB2LA0RB_1LA1RZ1RA 20 5
1RB0LB1RZ_2LA2LB1RA 19 7
1RB0RB1RZ_1LB2RA1LA 19 6
1RB2LB1LA_2LA1RZ2RB 18 7
1RB1LB2LA_1LA2RB1RZ 18 7
1RB2LB2LA_2LA1RZ0RA 18 6
1RB2LA1RZ_1LB1RA2RB 18 6
1RB2LB1RZ_2LA1RA0LB 18 5
1RB0RB1RZ_1LB2LA2RA 17 6
1RB2LB0LB_2LA1RZ2RB 17 4
1RB2RA1RZ_0LB2RB1LA 17 3

For the full list of 866 halting BB(2,3) TMs, see: https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/2x3.txt

==Sources==
<references />

[[Category:BB Domains]]

SKI Calculus

2026-05-10T21:43:41Z

C7X: Replace with archived version

A '''SKI calculus''' program is a binary tree where the leaves are combinators, the three symbols <code>S</code>, <code>K</code>, <code>I</code>. Using parentheses to notate the tree, a simple example of a SKI program is <code>(((SK)S)((KI)S))</code>. We can omit parentheses by assuming they are left-binding by default, so we simplify our program to <code>SKS(KIS)</code>.

Like lambda calculus, SKI calculus has a process called beta-reduction. We change the tree according to any reducible redex.

* <code>Ix -> I</code>
* <code>Kxy -> Kx</code>
* <code>Sxyz -> Sxz(yz)</code>

Note that <code>xyz</code> represent any valid trees, not just single combinators. We repeat this process and we say it terminates if the combinator cannot be beta-reduced.

Busy Beaver for SKI calculus ('''BB_SKI''') is a variation of the [[Busy Beaver for lambda calculus|Busy Beaver problem for lambda calculus]]. BB_SKI(n) is defined as the size of the largest output of a terminating program of size n.

== Champions ==
{| class="wikitable"
! n !! Value !! Champion !! Discoverered by
|-
| 1 || = 1 || S || ?
|-
| 2 || = 2 || SS || ?
|-
| 3 || = 3 || SSS || ?
|-
| 4 || = 4 || SSSS || ?
|-
| 5 || = 6 || SSS(SS) || ?
|-
| 6 || ≥ 17 || SSS(SI)S || ?
|-
|7
|≥ 40
|S(SS)S(SS)S
|?
|-
|8
|≥ 41
|SII(S(S(SS)))S
|?
|-
|9
|≥ 79
|SII(SS(SSS))S
|?
|-
|10
|≥ 164
|SII(SS(SS(SS)))S
|?
|-
|11
|≥ 681
|SII(SS(SS(SSS)))S
|?
|-
|12
|≥ 1530
|SII(SS(SS(SS(SS))))S
|?
|-
|13
|≥ 65537
|S(S(SI))I(S(S(KS)K)I)KK
|?
|-
|14
|≥ 2^256+1
|S(S(S(SI)))I(S(S(KS)K)I)KK
|?
|-
|15
|> 2^2^2^2^21
|S(S(SSS)I)I(S(S(KS)K)I)KK
|?
|-
|16
|> 2^^19
|S(S(S(SSS))I)I(S(S(KS)K)I)KK
|?
|-
|17
|> 2^^2^128
|SSK(S(S(KS)K)I)(S(SI(SI))I)KK
|?
|-
|18
|> 2^^2^2^2^2^21
|SSK(S(S(KS)K)I)(S(S(SSS)I)I)KK
|?
|-
|19
|> 2^^^2^128
|S(SSK(S(SI(SI))I))I(S(S(KS)K)I)KK
|?
|-
|20
|> 2^^^2^2^2^2^21
|S(SSK(S(S(SSS)I)I))I(S(S(KS)K)I)KK
|?
|-
|21
|> 2^^^2^^19
|S(SSK(S(S(S(SSS))I)I))I(S(S(KS)K)I)KK
|?
|-
|25
|> Graham's Number
|SII(SI(SI(K(S(K(S(K(SS(K(K(S(S(KS)K)I)))))(SI)))K))))
| 2014MELO03
|}

== SK calculus ==
We can remove the <code>I</code> combinator and replace it by <code>(SKS)</code>, <code>(SKK)</code> or any <code>(SKx)</code>. These terms have a straightforward binary encoding where (prefix) application is 1, K=00, and S=01. Since n combinators take n-1 applications to combine, their code length is 2n + n-1 = 3n-1 bits.

=== Champions ===
{| class="wikitable"
! n !! bits !! Value !! Champion !! Discoverered by
|-
| 1 || 2 || = 1 || S || ?
|-
| 2 || 5 || = 2 || SS || ?
|-
| 3 || 8 || = 3 || SSS || ?
|-
| 4 || 11 || = 4 || SSSS || ?
|-
| 5 || 14 || = 6 || SSS(SS) || ?
|-
| 6 || 17 || ≥ 10 || SSS(SS)S || ?
|-
|7
| 20
|≥ 40
|S(SS)S(SS)S
|?
|-
|8
| 23
|≥ 41
|S(S(SS)S(SS)S)
|?
|-
|9
| 26
|≥ 42
|S(S(S(SS)S(SS)S))
|?
|-
|10
| 29
|≥ 66
|SS(SSS)(SS(SS))S
|?
|-
|11
| 32
|≥ 79
|SS(SSS)(SS(SSS))S
|?
|-
|12
| 35
|≥ 164
|SS(SKK)(SS)(SS(SS))S
|?
|-
|13
| 38
|≥ 681
|SS(SKK)(SS)(SS(SSS))S
|?
|-
|14
| 41
|≥ 1530
|SS(SKK)(SS)(SS(SS(SS)))S
|?
|-
|15
| 44
|≥ 7811
|SS(SKK)(SS)(SS(SS(SSS)))S
|?
|}

== See Also ==
[https://web.archive.org/web/20250217071017/https://komiamiko.me/math/ordinals/2020/06/21/ski-numerals.html Lower bounds of this function] (archived)

[https://dallaylaen.github.io/ski-interpreter/ SKI interpreter]

[[Category:Functions]]

Fast-Growing Hierarchy

2026-04-22T19:26:39Z

C7X: Spacing /* Examples */

A '''Fast-Growing Hierarchy''' (FGH) is an ordinal-indexed hierarchy of functions satisfying certain restrictions. FGHs are used for assigning growth rates to fast computable functions, and are useful for approximating scores and halting times of [[Turing machine|Turing machines]].
[[Category:Functions]]
== Definition ==

A fundamental sequence for an ordinal <math>\alpha</math> is an increasing sequence of ordinals <math><\alpha</math> which is unbounded in <math>\alpha</math>. The <math>\beta</math>-th element of <math>\alpha</math>'s fundamental sequence is denoted by <math>\alpha[\beta]</math>. In the context of FGHs, there is usually a restriction that the sequence's length must be as small as possible (that is, the length is the [https://en.wikipedia.org/wiki/Cofinality cofinality] of <math>\alpha</math>). A system of fundamental sequences for a set of ordinals is a function which assigns a fundamental sequence to each ordinal in the set.

Given a system of fundamental sequences for limit ordinals below <math>\lambda</math>, its corresponding FGH is defined by

<math display="block">\begin{array}{l}
f_0(n) & = & n+1 \\
f_{\alpha+1}(n) & = & f_\alpha^n(n) & \text{for }\alpha<\lambda \\
f_\alpha(n) & = & f_{\alpha[n]}(n) & \text{for limit ordinals }\alpha<\lambda
\end{array}</math>

Most natural fundamental sequence systems almost exactly agree on the growth rates in their corresponding FGHs. Specifically, if <math>f,f'</math> are FGHs given by natural fundamental sequence systems, it is usually the case that <math>f_\alpha(n+1)>f'_\alpha(n)</math> and <math>f'_\alpha(n+1)>f_\alpha(n)</math> for all natural <math>n</math> and all successor ordinals <math>\alpha</math>. For this reason, the specific choice of a fundamental sequence system often doesn't matter for large ordinals. For small ordinals (below <math>\varepsilon_0</math>), a common choice of fundamental sequences is given by

<math display="block">\begin{array}{l}
(\alpha+\omega^{\beta+1})[n] & = & \alpha+\omega^\beta n & \text{if }\alpha\text{ is a multiple of }\omega^{\beta+1} \\
(\alpha+\omega^\beta)[n] & = & \alpha+\omega^{\beta[n]} & \text{if }\alpha\text{ is a multiple of }\omega^\beta\text{ and }\beta\text{ is a limit ordinal}
\end{array}</math>

The FGH given by these fundamental sequences is sometimes called the Wainer hierarchy. Above <math>\varepsilon_0</math>, a relatively elegant choice is the expansion associated to the [https://apeirology.com/wiki/Bashicu_matrix_system Bashicu matrix system], which has the [https://en.wikipedia.org/wiki/Fundamental_sequence_(set_theory)#Additional_conditions Bachmann property].

== Examples ==
* <math>f_\alpha(0) = 0</math> for <math>\alpha > 0</math>
* <math>f_\alpha(1) = 2</math>

* <math>f_0(n) = n+1</math>
* <math>f_1(n) = 2n</math>
* <math>f_2(n) = n \cdot 2^n</math>
* <math>f_3(n) = f_2^n(n) > (2 \uparrow)^n(n + \log_2 n) > (2 \uparrow)^n n \ge 2 \uparrow\uparrow (n+1)</math> for <math>n \ge 2</math>
* <math>f_3(n) > 10 \uparrow\uparrow n</math> for <math>n \ge 4</math>
* <math>f_k(n) > 10 \uparrow^{k-1} n \; \text{ for } k \ge 4 \text{ and } n \ge 2 </math>
* <math>f_\omega(n) = f_n(n) > 10 \uparrow^{n-1} n \; \text{ for } n \ge 4</math>
* <math>f_{\omega+1}(64) = f_\omega^{64}(64) > \text{Graham's number}</math>

*

=== Table of Small Values ===
{| class="wikitable"
|+
!
! colspan="8" |n
|-
!
!0
!1
!2
!3
!4
!5
!...
!n
|-
|<math>f_0(n)</math>
|1
|2
|3
|4
|5
|6
|
|n+1
|-
|<math>f_1(n)</math>
|0
|2
|4
|6
|8
|10
|
|2n
|-
|<math>f_2(n)</math>
|0
|2
|8
|24
|64
|160
|
|<math>n \, 2^n</math>
|-
|<math>f_3(n)</math>
|0
|2
|2048
|<math>>10^{10^8}</math>
|<math>> 10^{10^{10^{20}}}</math>
|<math>> (10 \uparrow)^4 49 > 10 \uparrow\uparrow 5</math>
|
|<math>> 10 \uparrow\uparrow n</math>
|-
|<math>f_4(n)</math>
|0
|2
|<math>> 10 \uparrow\uparrow 2048</math>
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^8} > 10 \uparrow\uparrow\uparrow 3</math>
|<math>> 10 \uparrow\uparrow\uparrow 4</math>
|<math>> 10 \uparrow\uparrow\uparrow 5</math>
|
|<math>> 10 \uparrow\uparrow\uparrow n</math>
|-
|...
|
|
|
|
|
|
|
|
|-
|<math>f_\omega(n)</math>
|1
|2
|8
|<math>>10^{10^8}</math>
|<math>> 10 \uparrow\uparrow\uparrow 4</math>
|<math>> 10 \uparrow^4 5</math>
|
|<math>> 10 \uparrow^{n-1} n</math>
|-
|<math>f_{\omega+1}(n)</math>
|0
|2
|<math>> 10 \uparrow^7 8</math>
|
|
|
|
|
|}

== Growth Bound Theorem ==
The ''Fast-Growing Hierarchy Growth Bound Theorem'' is an important result in mathematical logic that has significant implications for unprovability results. The theorem highlights a relationship between computable functions that are provably total in first-order Peano Arithmetic (PA) and the fast-growing functions in the [[Fast-Growing Hierarchy|Wainer hierarchy]].

The theorem is based on work by several mathematicians. Georg Kreisel laid the groundwork in 1952 by investigating connections between recursions over well ordered sets and proofs in PA. These results were subsequently extended by many others; the following form is based on the presentation by Buchholz and Wainer.

=== Statement ===
Let <math>T</math> be a Turing machine that computes a function <math>g:\N\to\N</math>, terminating on every input. Suppose that PA can prove the statement «<math>T</math> terminates on every input.» Then <math>g</math> cannot grow too fast: There exist <math>\alpha < \varepsilon_0</math> and <math>n_0\in\N</math> such that <math>g(n) < F_\alpha(n)</math> for every <math>n\ge n_0</math>.

== References ==
Wilfried Buchholz and Stan S. Wainer. Provably computable functions and the fast growing hierarchy. In S. G. Simpson, editor, Logic and Combinatorics, volume 65 of Contemporary Mathematics, pages 179–198. AMS, 1987. [https://epub.ub.uni-muenchen.de/3843/1/3843.pdf]

Fast-Growing Hierarchy

2026-04-22T19:26:13Z

C7X: f_ω(0) = f_0(0) = 1 /* Table of Small Values */

A '''Fast-Growing Hierarchy''' (FGH) is an ordinal-indexed hierarchy of functions satisfying certain restrictions. FGHs are used for assigning growth rates to fast computable functions, and are useful for approximating scores and halting times of [[Turing machine|Turing machines]].
[[Category:Functions]]
== Definition ==

A fundamental sequence for an ordinal <math>\alpha</math> is an increasing sequence of ordinals <math><\alpha</math> which is unbounded in <math>\alpha</math>. The <math>\beta</math>-th element of <math>\alpha</math>'s fundamental sequence is denoted by <math>\alpha[\beta]</math>. In the context of FGHs, there is usually a restriction that the sequence's length must be as small as possible (that is, the length is the [https://en.wikipedia.org/wiki/Cofinality cofinality] of <math>\alpha</math>). A system of fundamental sequences for a set of ordinals is a function which assigns a fundamental sequence to each ordinal in the set.

Given a system of fundamental sequences for limit ordinals below <math>\lambda</math>, its corresponding FGH is defined by

<math display="block">\begin{array}{l}
f_0(n) & = & n+1 \\
f_{\alpha+1}(n) & = & f_\alpha^n(n) & \text{for }\alpha<\lambda \\
f_\alpha(n) & = & f_{\alpha[n]}(n) & \text{for limit ordinals }\alpha<\lambda
\end{array}</math>

Most natural fundamental sequence systems almost exactly agree on the growth rates in their corresponding FGHs. Specifically, if <math>f,f'</math> are FGHs given by natural fundamental sequence systems, it is usually the case that <math>f_\alpha(n+1)>f'_\alpha(n)</math> and <math>f'_\alpha(n+1)>f_\alpha(n)</math> for all natural <math>n</math> and all successor ordinals <math>\alpha</math>. For this reason, the specific choice of a fundamental sequence system often doesn't matter for large ordinals. For small ordinals (below <math>\varepsilon_0</math>), a common choice of fundamental sequences is given by

<math display="block">\begin{array}{l}
(\alpha+\omega^{\beta+1})[n] & = & \alpha+\omega^\beta n & \text{if }\alpha\text{ is a multiple of }\omega^{\beta+1} \\
(\alpha+\omega^\beta)[n] & = & \alpha+\omega^{\beta[n]} & \text{if }\alpha\text{ is a multiple of }\omega^\beta\text{ and }\beta\text{ is a limit ordinal}
\end{array}</math>

The FGH given by these fundamental sequences is sometimes called the Wainer hierarchy. Above <math>\varepsilon_0</math>, a relatively elegant choice is the expansion associated to the [https://apeirology.com/wiki/Bashicu_matrix_system Bashicu matrix system], which has the [https://en.wikipedia.org/wiki/Fundamental_sequence_(set_theory)#Additional_conditions Bachmann property].

== Examples ==
* <math>f_\alpha(0) = 0</math>for <math>\alpha > 0</math>
* <math>f_\alpha(1) = 2</math>

* <math>f_0(n) = n+1</math>
* <math>f_1(n) = 2n</math>
* <math>f_2(n) = n \cdot 2^n</math>
* <math>f_3(n) = f_2^n(n) > (2 \uparrow)^n(n + \log_2 n) > (2 \uparrow)^n n \ge 2 \uparrow\uparrow (n+1)</math> for <math>n \ge 2</math>
* <math>f_3(n) > 10 \uparrow\uparrow n</math> for <math>n \ge 4</math>
* <math>f_k(n) > 10 \uparrow^{k-1} n \; \text{ for } k \ge 4 \text{ and } n \ge 2 </math>
* <math>f_\omega(n) = f_n(n) > 10 \uparrow^{n-1} n \; \text{ for } n \ge 4</math>
* <math>f_{\omega+1}(64) = f_\omega^{64}(64) > \text{Graham's number}</math>

*

=== Table of Small Values ===
{| class="wikitable"
|+
!
! colspan="8" |n
|-
!
!0
!1
!2
!3
!4
!5
!...
!n
|-
|<math>f_0(n)</math>
|1
|2
|3
|4
|5
|6
|
|n+1
|-
|<math>f_1(n)</math>
|0
|2
|4
|6
|8
|10
|
|2n
|-
|<math>f_2(n)</math>
|0
|2
|8
|24
|64
|160
|
|<math>n \, 2^n</math>
|-
|<math>f_3(n)</math>
|0
|2
|2048
|<math>>10^{10^8}</math>
|<math>> 10^{10^{10^{20}}}</math>
|<math>> (10 \uparrow)^4 49 > 10 \uparrow\uparrow 5</math>
|
|<math>> 10 \uparrow\uparrow n</math>
|-
|<math>f_4(n)</math>
|0
|2
|<math>> 10 \uparrow\uparrow 2048</math>
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^8} > 10 \uparrow\uparrow\uparrow 3</math>
|<math>> 10 \uparrow\uparrow\uparrow 4</math>
|<math>> 10 \uparrow\uparrow\uparrow 5</math>
|
|<math>> 10 \uparrow\uparrow\uparrow n</math>
|-
|...
|
|
|
|
|
|
|
|
|-
|<math>f_\omega(n)</math>
|1
|2
|8
|<math>>10^{10^8}</math>
|<math>> 10 \uparrow\uparrow\uparrow 4</math>
|<math>> 10 \uparrow^4 5</math>
|
|<math>> 10 \uparrow^{n-1} n</math>
|-
|<math>f_{\omega+1}(n)</math>
|0
|2
|<math>> 10 \uparrow^7 8</math>
|
|
|
|
|
|}

== Growth Bound Theorem ==
The ''Fast-Growing Hierarchy Growth Bound Theorem'' is an important result in mathematical logic that has significant implications for unprovability results. The theorem highlights a relationship between computable functions that are provably total in first-order Peano Arithmetic (PA) and the fast-growing functions in the [[Fast-Growing Hierarchy|Wainer hierarchy]].

The theorem is based on work by several mathematicians. Georg Kreisel laid the groundwork in 1952 by investigating connections between recursions over well ordered sets and proofs in PA. These results were subsequently extended by many others; the following form is based on the presentation by Buchholz and Wainer.

=== Statement ===
Let <math>T</math> be a Turing machine that computes a function <math>g:\N\to\N</math>, terminating on every input. Suppose that PA can prove the statement «<math>T</math> terminates on every input.» Then <math>g</math> cannot grow too fast: There exist <math>\alpha < \varepsilon_0</math> and <math>n_0\in\N</math> such that <math>g(n) < F_\alpha(n)</math> for every <math>n\ge n_0</math>.

== References ==
Wilfried Buchholz and Stan S. Wainer. Provably computable functions and the fast growing hierarchy. In S. G. Simpson, editor, Logic and Combinatorics, volume 65 of Contemporary Mathematics, pages 179–198. AMS, 1987. [https://epub.ub.uni-muenchen.de/3843/1/3843.pdf]

Fractran

2026-04-12T19:36:00Z

C7X: Name of cryptid /* Visualizing Fractran Programs' Space-Time Diagrams */

'''Fractran''' (originally styled FRACTRAN) is an esoteric [[Turing complete]] model of computation invented by John Conway in 1987.<ref>Conway, John H. (1987). "FRACTRAN: A Simple Universal Programming Language for Arithmetic". ''Open Problems in Communication and Computation''. Springer-Verlag New York, Inc. pp. 4–26. <nowiki>http://doi.org/10.1007/978-1-4612-4808-8_2</nowiki></ref> In this model a program is simply a finite list of fractions (rational numbers), the program state is an integer. For more details see https://en.wikipedia.org/wiki/FRACTRAN.

Discord user Coda came up with a way to transform any Fractran program into a Turing Machine, see [https://discord.com/channels/960643023006490684/1438019511155691521/1441844795613122560 source].

'''BB_fractran'''(n) or '''BBf'''(n) is the Busy Beaver function for Fractran programs. Holdouts lists by Daniel Yuan: [https://github.com/int-y1/BBFractran/blob/main/holdout/README.md Holdouts lists]

== Definition ==
A Fractran program is a list of rational numbers <math>[q_0, q_1, \dots, q_{k-1}]</math> called rules and a Fractran state is an integer <math>s \in \mathbb{Z}</math>. The numerator and denominator of any rational number fraction do not share any prime factors (they are in reduced form). We say that a rule <math>q_i</math> applies to state <math>s</math> if <math>s \cdot q_i \in \mathbb{Z}</math>. If no rule applies, we say that the computation has halted otherwise we apply the first applicable rule at each step. In that case we say <math>s \to t</math> and <math>t = s \cdot q_i</math> and <math>i = \min \{ i : s \cdot q_i \in \mathbb{Z} \}</math>. As with [[Turing machines]], we will write <math>s \xrightarrow{N} t</math> if <math>s \to s_1 \to \cdots \to s_{N-1} \to t</math> (s goes to t after N steps) and <math>s \to^* t</math> or <math>s \to^+ t</math> if <math>s \xrightarrow{N} t</math> for some N≥0 or N≥1 (respectively). We say that a program has runtime N (or halts in N steps) starting in state s if <math>s \xrightarrow{N} t</math> and computation halts on t.

Let <math>\Omega(n)</math> be the total number of prime factors of a positive integer n. In other words, <math>\Omega(2^{a_0} 3^{a_1} \cdots p_n^{a_n}) = \sum_{k=0}^n a_n</math>. Then given a rule <math>\frac{a}{b}</math> we say that <math>\text{size} \left( \frac{a}{b} \right) = \Omega(a) + \Omega(b)</math>. And the size of a Fractran program <math>[q_0, q_1, \dots, q_{k-1}]</math> is <math>k + \sum_{i=0}^{k-1} \text{size}(q_i)</math>.

BB_fractran(n) or BBf(n) is the maximum runtime starting in state 2 for all halting Fractran programs of size n. It is a non-computable function akin to the [[Busy Beaver Functions]] since Fractran is Turing Complete.

== Vector Representation ==
Fractran programs are not easy to interpret, in fact it may be completely unclear at first that they can perform any computation at all. One of the key insights is to represent all numbers (states and rules) in their prime factorization form. For example, we can use a vector <math>[ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{Z}^n</math> to represent the number <math>2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math>.

Let the vector representation (for a sufficiently large n) for a state <math>a = 2^{a_0} 3^{a_1} \cdots p_{n-1}^{a_{n-1}}</math> be <math>v(a) = [ a_0, a_1, \dots, a_{n-1} ] \in \mathbb{N}^n</math> and the vector representation for a rule <math>\frac{a}{b}</math> be <math>v \left( \frac{a}{b} \right) = v(a) - v(b) \in \mathbb{Z}^n</math> (Note that this is just an extension of the original definition extended to allow negative <math>a_i</math>).

Now, rule q applies to state s iff <math>v(s) + v(q) \in \mathbb{N}^n</math> (all components of the vector are ≥0) and if <math>s \to t</math> then <math>v(t) = v(s) + v(q)</math>. So the Fractran multiplication model is completely equivalent to the vector adding model. For presentation, we will represent a Fractran program with a matrix where each row is the vector representation for a rule.

For example, the BBf(15) champion (<code>[1/45, 4/5, 3/2, 25/3]</code>) in vector representation would be:

<math display="block">\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>

In this representation, it becomes much easier to reason about Fractran programs and describe general rules. It is also very easy to calculate the size of a rule or program in vector representation. It is the sum of absolute values of all elements in the matrix + number of rules (number of rows).

=== Relationship to VAS / Petri Nets ===
Using vector representation, Fractran programs are a deterministic version of [[wikipedia:Vector_addition_system|Vector Addition Systems (VAS)]] (and, equivalently, [[wikipedia:Petri_net|Petri Nets]]). VAS are identical to Fractran programs in vector representation except that the rules are unordered and non-deterministic, they are used to model distributed systems where precise order of rule execution cannot be predicted. Interestingly, many problems about VAS are actually decidable, but their runtimes are extremely slow. Notably, the reachability problem (given states A and B are there a sequence of rules so that <math>A \to^* B</math>) is "Ackermann-complete" meaning that the optimal algorithm has worst-case runtime akin to the famously fast-growing Ackermann function.<ref>Czerwiński, Wojciech; Orlikowski, Łukasz (2021). ''Reachability in Vector Addition Systems is Ackermann-complete''. 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS). https://arxiv.org/abs/2104.13866.</ref>

== Visualizing Fractran Programs' Space-Time Diagrams ==
Katelyn Doucette's Fractran space-time diagram visualizer produces the following space-time diagrams for some notable Fractran Programs, under the following principle: Each color represents a prime factor. Left -> right colors indicating the index of that register, and how wide the color is representing how big the value is at that step. Source code: https://github.com/Laturas/FractranVisualizer
{| class="wikitable"
|+
|[[File:Fractran_22_Cryptid.webp|alt=The space-time diagram of Fenrir|460x460px]]
The space-time diagram of Fenrir
|[[File:Hydra.webp|alt=The space-time diagram of Hydra.|460x460px]]
The space-time diagram of Hydra.
|[[File:Bbf21 champ full.png|alt=The space-time diagram of the BBf(21) champion.|400x400px]]

The space-time diagram of the BBf(21) champion. The width & height of the diagram can be set in the visualizer.
|[[File:Space_Needle.webp|alt=The space-time diagram of Space Needle.|460x460px]]
The space-time diagram of Space Needle.
|}

== Deciders ==
[[File:Fractran deciders.png|alt=Fractran deciders|thumb|All Fractran deciders summarized and their relations, shared by Daniel Yuan on [https://discord.com/channels/960643023006490684/1438019511155691521/1439001835904958655 14 Nov 2025]]]Many specialized deciders have been invented to prove Fractran programs non-halting. See image at right. There are three extra deciders: [https://discord.com/channels/960643023006490684/1438019511155691521/1449775657554022531 Spanning Vectors Masked,] which should be very effective, but implementing it is in-progress, a version of Spanning Vectors Masked - [https://discord.com/channels/960643023006490684/1438019511155691521/1453217977385091092 Masked Linear Invariant] - which is very powerful, and some holdouts were removed by [[User:Sligocki|Shawn Ligocki]] with [https://lsv.ens-paris-saclay.fr/Software/fast/ FAST] (Fast Acceleration of Symbolic Transition systems), a pre-existing general tool.

-d released a new decider on 25 Jan 2026: [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 Beeping Permutation].

== Champions ==
The table of champions is split into two pieces: the first for small champions (up to BBf(14)) which all share the same relatively simple behavior (sequential programs) is collapsed by default; the second for champions BBf(15) and beyond which have more complex and varied behavior.
All small champions as well as the first few larger ones were discovered and proven maximal by Jason Yuen (@-d) in their initial enumeration on [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025].

<div class="toccolours mw-collapsible mw-collapsed">'''Small Champions'''<div class="mw-collapsible-content">
{| 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>
|}
</div></div>

{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
!Champion Found
!Holdouts Proven
|-
| 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>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 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>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 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>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434313398799175710 1 Nov 2025]
|Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1362008236118511758/1434771877376557086 3 Nov 2025]
|-
| 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>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1435313806493614131 4 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1436661215911870584 8 Nov 2025]
|-
| 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>
|@creeperman7002 [https://discord.com/channels/960643023006490684/1362008236118511758/1435763150489387090 5 Nov 2025]
|Decider: Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438558242388312165 13 Nov 2025]
3 Holdouts: Racheline & Shawn Ligocki
|-
|20
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1438480761169776733 13 Nov 2025]
|Decider: Jason Yuen (@-d)
([https://github.com/int-y1/BBFractran/tree/main/holdout Enum+initial])
Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438559507579011194 13] and [https://discord.com/channels/960643023006490684/1438019511155691521/1438996636389998773 14 Nov 2025]

Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1447069110541484146 7] and [https://discord.com/channels/960643023006490684/1438019511155691521/1453213088630444168 24 Dec 2025]
6 Holdouts: Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1452913055053778945 23 Dec 2025]
|-
|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>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1439759182587891894 16 Nov 2025]
|140 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 25 Jan 2026]
Claude Opus 4.6 proof of nonhalting of all 140: [https://discord.com/channels/960643023006490684/1438019511155691521/1485168251997786173 28 March 2026]
|-
|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>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1448912286713384961 11 Dec 2025] and Jason Yuen (@-d)<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1448953682237460480 <nowiki>[1]</nowiki>]</sup>
|2003 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 25 Jan 2026]

Known [[Cryptid|Cryptids]]:

# Fenrir
|-
|23
|
|
|
|
|Known [[Cryptid|Cryptids]]:

# Frankenstein's Monster
# Antihydra-like Cryptid
|}

=== Behavior of Champions ===

==== Sequential programs ====
All champions up to BBf(14) have very simple behavior. They are all of the form: <math>\left[ \frac{3^{a_1}}{2}, \frac{5^{a_2}}{3}, \dots, \frac{p_n^{a_k}}{p_{k-1}}, \frac{1}{p_k} \right]</math> or in vector representation (limited to k=4):

<math display="block">\begin{bmatrix}
-1 & a_1 & 0 & 0 & 0 \\
0 & -1 & a_2 & 0 & 0 \\
0 & 0 & -1 & a_3 & 0 \\
0 & 0 & 0 & -1 & a_4 \\
0 & 0 & 0 & 0 & -1
\end{bmatrix}</math>

These champions repeatedly apply the rules in sequence, never going back to a previous rule. They apply the first rule until they've exhausted all 2s, then the second rule until they've exhausted all 3s, etc. They have a runtime of <math>1 + a_1 + a_1 a_2 + a_1 a_2 a_3 + \cdots = \sum_{i=0}^k \prod_{j=1}^i a_j</math> and size <math>2k+2 + \sum_{i=1}^k a_i</math>. This grows linearly for k=1 (BBf(5) to BBf(10)) and quadratically for k=2 (BBf(11) to BBf(14)). Letting k grow with the size, the maximum runtime grows exponentially in the program size.

==== BBf(15) Family ====
The BBf(15) and BBf(16) champions are members of a family of programs (parameterized by <math>n \ge 1</math>):

<math display="block">\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & n
\end{bmatrix}</math>

Let a = 2, b = 3, and c = 5.

The BBf(15) champion (n = 2) implements this iteration:

<math display="block">\begin{array}{lcl}
b^0 & \xrightarrow{0} & \text{halt} \\
b^1 & \xrightarrow{7} & b^4 \\
b^2 & \xrightarrow{7} & b^5 \\
b^3 & \xrightarrow{5} & b^2 \\
b^4 & \xrightarrow{5} & b^3 \\
b^{n+5} & \xrightarrow{3} & b^n \\
\end{array}</math>

which follows a permutation-like trajectory: <math>a \xrightarrow{1} b^1 \to b^4 \to b^3 \to b^2 \to b^5 \to b^0 \to \text{halt}</math>

The BBf(16) champion (n = 3) implements this iteration:

<math display="block">\begin{array}{lcl}
b^0 & \xrightarrow{0} & \text{halt} \\
b^1 & \xrightarrow{10} & b^6 \\
b^2 & \xrightarrow{10} & b^7 \\
b^3 & \xrightarrow{8} & b^4 \\
b^4 & \xrightarrow{8} & b^5 \\
b^5 & \xrightarrow{6} & b^2 \\
b^6 & \xrightarrow{6} & b^3 \\
b^{n+7} & \xrightarrow{4} & b^n \\
\end{array}</math>

which follows a permutation-like trajectory: <math>a \xrightarrow{1} b^1 \to b^6 \to b^3 \to b^4 \to b^5 \to b^2 \to b^7 \to b^0 \to \text{halt}</math>

==== BBf(17) Family ====
The BBf(17) to BBf(19) champions are members of a family of programs (parameterized by <math>m,n \ge 0</math>)

<math display="block">\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & n \\
0 & 1 & -1 & 0 \\
m & 0 & 1 & -1
\end{bmatrix}</math>

which have size <math>m+n+12</math>

This family obeys the following rules:

# <math>[1, 0, 0, 0] \xrightarrow{1} [0, 0, 0, n]</math>
# if d≥1 and b≤m:<math display="block">[0, b, 0, d] \xrightarrow{m+b+2} [0, b+1, 0, d - 1 + n(m-b)]</math>
# if d≥1 and b≥m:<math display="block">[0, b, 0, d] \xrightarrow{2m+2} [0, b+1, 0, d - 1]</math>
#if d=0: [0,b,0,d] has halted

and furthermore these rules are applied in order since b is always increasing (and d is eventually decreasing). Combining these together we get runtime:<math display="block">1 + n(m+1)(m(m+1)+2) - \frac{m(m+1)}{2}</math>

The optimal choices for n,m for various program sizes are:
{| class="wikitable"
|+
!Size
!n
!m
!Runtime
|-
|16
|1
|3
|51
|-
|'''''17'''''
|'''''2'''''
|'''''3'''''
|'''''107'''''
|-
|'''''18'''''
|'''''2'''''
|'''''4'''''
|'''''211'''''
|-
|'''''19'''''
|'''''2'''''
|'''''5'''''
|'''''370'''''
|-
|20
|2
|6
|596
|-
|21
|3
|6
|904
|}

<br>
==== BBf(20) ====
[[File:Screenshot 2026-04-01 104704.png|alt=Full space-time diagram of the BBf(20) champion.|left|507x507px]]
The BBf(20) champion (running 746 steps):

<math display="block">\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>

This program implements a [[Collatz-like]] iteration. Let <math>C(n) = [0, 0, n, 2, 0]</math>, then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{49} & C(2) \\
C(3k) & \xrightarrow{3k} & \text{halt} \\
C(3k+1) & \xrightarrow{11k+22} & C(4k+3) \\
C(3k+2) & \xrightarrow{11k+22} & C(4k+4) \\
\end{array}</math>

which follows the reasonably "lucky" trajectory:

<math display="block">C(2) \to C(4) \to C(7) \to C(11) \to C(16) \to C(23) \to C(32) \to C(44) \to C(60) \to \text{halt}</math>

==== BBf(21) ====
[[File:Bbf21 champ full.png|alt=The full space-time diagram of the BBf(21) champion until halting.|thumb|The full space-time diagram of the BBf(21) champion until halting.]]

The BBf(21) champion (running >31M steps):
<math display="block">\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>

This program implements a Collatz-like iteration. Let <math>D(n) = [0, 0, n, 0]</math>, then:<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1439779341365022852]</sup>
<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & D(1) \\
D(3k) & \xrightarrow{k} & \text{halt} \\
D(3k+1) & \xrightarrow{21k+7} & C(10k+4) \\
D(3k+2) & \xrightarrow{21k+14} & C(10k+7) \\
\end{array}</math>

which follows the reasonably "lucky" trajectory:
<math display="block">\begin{array}{ll}
D(1) & \to D(4) \to D(14) \to D(47) \to D(157) \to D(524) \to D(1747) \to D(5824) \to D(19414) \\
& \to D(64714) \to D(215714) \to D(719047) \to D(2396824) \to D(7989414) \to \text{halt} \\
\end{array}</math>

==== BBf(22) ====
The BBf(22) champion (running <math>> 10^{62}</math> steps):
<math display="block">\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>

This program implements a [[Collatz-like]] unbiased pseudo-random walk. Let <math>S(x,y) = [0, 0, x, 0, y]</math>, then:<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1449118888142049421]</sup>
<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & S(0,1) \\
S(x, 0) & = & \text{halt} \\
S(3k, y+1) & \xrightarrow{14k+4} & S(5k+1, y+1) \\
S(3k+1, y+1) & \xrightarrow{14k+10} & S(5k+3, y+2) \\
S(3k+2, y+1) & \xrightarrow{14k+12} & S(5k+4, y) \\
\end{array}</math>

This random walk iterates 275 times until it halts reaching a maximum y value of 14 at iteration 111:
<math display="block">\begin{array}{ll}
S(0,1) & \to S(1,1) \to S(3,2) \to S(6,2) \to S(11, 2) \to S(19, 1) \to S(33, 2) \to S(56, 2) \to S(94, 1) \\
& \to S(158, 2) \to S(264, 1) \to S(441, 1) \to S(736, 1) \to S(1228, 2) \to S(2048, 3) \\
& \vdots \\
& \to S(4065328691604230522442358, 13) \\
& \to S(6775547819340384204070598, 14) \\
& \to S(11292579698900640340117664, 13) \\
& \vdots \\
& \to S(27930059557111373800280446055462487109112535227834136644, 2) \\
& \to S(46550099261852289667134076759104145181854225379723561074, 1) \\
& \to S(77583498769753816111890127931840241969757042299539268458, 2) \\
& \to S(129305831282923026853150213219733736616261737165898780764, 1) \\
& \to S(215509718804871711421917022032889561027102895276497967941, 1) \\
& \to S(359182864674786185703195036721482601711838158794163279903, 2) \\
& \vdots \\
& \to S(5894430516013404355095519889620117404469367857588232386361874, 2) \\
& \to S(9824050860022340591825866482700195674115613095980387310603124, 1) \\
& \to S(16373418100037234319709777471166992790192688493300645517671874, 0)
\end{array}</math>

== Cryptids ==

=== Fenrir ===
[[File:Fractran 22 Cryptid.webp|alt=The space-time diagram of Fenrir.|thumb|Partial space-time diagram of Fenrir.]]
"Fenrir" is a family of 3 size 22 [[Cryptids]] discovered by Jason Yuen (@-d) and Claude Opus 4.6 on 22 Mar 2026. Out of 500 holdouts of size 22, Claude Opus 4.6 used Lean to prove that 497 holdouts were non-halting. The remaining 3 holdouts are the Fenrir family.<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1485415054475268179]</sup> Discord user @ZTS439 shared [https://discord.com/channels/960643023006490684/1438019511155691521/1487251919444508723 some analysis] and a [https://discord.com/channels/960643023006490684/1438019511155691521/1487252789158613002 Python program] for it. Its name comes from [[wikipedia:Norse_mythology|nordic mythology]]; [[wikipedia:Fenrir|Fenrir]] is the wolf that helps destroy the world during [[wikipedia:Ragnarök|Ragnarök]].

{| class="wikitable"
|+
!Holdout number
!Holdout
!Vector Representation
|-
| 29/2003
| <code>[1/15, 27/77, 49/3, 10/49, 33/2]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 0 & 0 \\
0 & 3 & 0 & -1 & -1 \\
0 & -1 & 0 & 2 & 0 \\
1 & 0 & 1 & -2 & 0 \\
-1 & 1 & 0 & 0 & 1
\end{bmatrix}</math>
|-
| 41/2003
| <code>[1/15, 49/3, 27/77, 10/49, 33/2]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 0 & 0 \\
0 & -1 & 0 & 2 & 0 \\
0 & 3 & 0 & -1 & -1 \\
1 & 0 & 1 & -2 & 0 \\
-1 & 1 & 0 & 0 & 1
\end{bmatrix}</math>
|-
| 430/2003
| <code>[27/35, 1/33, 25/3, 22/25, 21/2]</code>
|<math>\begin{bmatrix}
0 & 3 & -1 & -1 & 0 \\
0 & -1 & 0 & 0 & -1 \\
0 & -1 & 2 & 0 & 0 \\
1 & 0 & -2 & 0 & 1 \\
-1 & 1 & 0 & 1 & 0
\end{bmatrix}</math>
|}

All 3 holdouts follow a biased random walk that somewhat resembles [[Hydra]]. Let <math>S(x,y) = [x, 0, 0, 2, y]</math> (for 29/2003 and 41/2003) or <math>S(x,y) = [x, 0, 2, y, 0]</math> (for 430/2003), then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \to & S(0,1) \\
S(0, 2y) & = & \text{halt} \\
S(x, 2y) & \to & S(x-1, 5y+2) \\
S(x, 2y+1) & \to & S(x+2, 5y)
\end{array}</math>

The first few visited states are $$S(0, 1) \to S(2, 0) \to S(1, 2) \to S(0, 7) \to S(2, 15) \to S(4, 35)$$

=== Frankenstein's Monster ===
[[File:Frankenstein's Monster.webp|alt=Partial space-time diagram of Frankenstein's Monster.|thumb|Partial space-time diagram of Frankenstein's Monster.]]
"Frankenstein's Monster" is a size 23 [[Cryptid]]. It was created by tweaking a single instruction in the size 22 champion. This tweak switches it from a unbiased random walk to a biased one and thus makes halting probviously impossible. It is called Frankenstein's Monster since it was found by a combination of exhaustive search and hand design.<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1449138938215141478]</sup>

<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code> <math display="block">\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 2 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>

Its behavior is extremely similar to the size 22 champion. Let <math>S(x,y) = [0, 0, x, 0, y]</math>, then:

<math display="block">\begin{array}{lcl}
[1,0,0,0,0] & \xrightarrow{1} & S(0,2) \\
S(x, 0) & = & \text{halt} \\
S(3k, y+1) & \xrightarrow{14k+4} & S(5k+1, y+2) \\
S(3k+1, y+1) & \xrightarrow{14k+10} & S(5k+3, y+4) \\
S(3k+2, y+1) & \xrightarrow{14k+12} & S(5k+4, y)
\end{array}</math>

with the only difference that the y values now change by {+1,+3,-1} depending on the value of x mod 3 (instead of {0,+1,-1} in the original size 22 program). The x values follow the exact same path as in the original size 22 champion, but the y values quickly grow linearly with the number of iterations (as expected by the random model):
0: S(0, 1) @ 1 (0.00s)
100_000: S(10^22_185, 100171) @ 10^22_186 (0.87s)
200_000: S(10^44_370, 200187) @ 10^44_371 (3.42s)
300_000: S(10^66_555, 300759) @ 10^66_556 (7.68s)
400_000: S(10^88_740, 400451) @ 10^88_741 (13.64s)
500_000: S(10^110_925, 500421) @ 10^110_925 (21.28s)
600_000: S(10^133_109, 600351) @ 10^133_110 (30.62s)
700_000: S(10^155_294, 700319) @ 10^155_295 (41.64s)
800_000: S(10^177_479, 799911) @ 10^177_480 (54.30s)
900_000: S(10^199_664, 900259) @ 10^199_665 (68.59s)
1_000_000: S(10^221_849, 1000853) @ 10^221_850 (84.51s)
...
4_000_000: S(10^887_395, 4000201) @ 10^887_396 (1474.02s)
...
27_500_000: S(10^6_100_841, 27512703) @ 10^6_100_842 (87616.45s)

=== Antihydra-like Cryptid ===
This Cryptid is a size 23 [[Cryptid]]. This Cryptid was [https://discord.com/channels/960643023006490684/1438019511155691521/1449293536737361973 constructed by Maksandchael] by tweaking Frankenstein's Monster to make it as similar to [[Antihydra]] as possible. <code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>

<math display="block">\begin{bmatrix}
-1 & 2 & -1 & 0 & 0 \\
-1 & -1 & 0 & 0 & 0 \\
-1 & 0 & 0 & 0 & 3 \\
1 & -1 & 0 & 1 & 0 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math><pre>
H(a, b) = [0, 0, a-2, 0, b]
Start -> H(2, 3)
H(2a, b) -> H(3a, b+2)
H(2a+1, b+1) -> H(3a+1, b)
H(a,0) -> halt
</pre>

=== Hydra ===
[[File:Hydra.webp|alt=Partial space-time diagram of Hydra.|thumb|300x300px|Partial space-time diagram of Hydra.]]
A size 25 program was produced and golfed by hand to simulate [[Hydra]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1449829146040467681 Discord]):

<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
<math display="block">
\begin{bmatrix}
-1 & 1 & 0 & -1 & 2 \\
-1 & 0 & 3 & 0 & 0 \\
1 & -1 & 0 & -1 & 1 \\
0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & -1 \\
1 & 0 & -1 & 1 & 0
\end{bmatrix}
</math>

The intended interpretation is that if we let <math>S(h,w) = [1, 0, w, h-3, 0]
</math> then it follows the following rules:

<math display="block">\begin{array}{lcl}
[1,0,\dots] & = & S(3, 0) \\
S(2k, 0) & \to^* & \text{halt} \\
S(2k, w+1) & \to^* & S(3k, w) \\
S(2k+1, w) & \to^* & S(3k+1, w+2)
\end{array}</math>

=== BMO1 ===
[[File:Ftran bmo1.png|alt=Partial space-time diagram of BMO 1.|thumb|Partial space-time diagram of BMO 1.]]
A size 36 program was produced by hand to simulate [[BMO1]] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1440018895212642424 Discord]):

<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
<math display="block">
\begin{bmatrix}
0 & 2 & -1 & 0 & -1 & 0 & 1 \\
1 & 0 & 0 & 0 & -1 & 0 & 0 \\
1 & 0 & -1 & -1 & 0 & 1 & 0 \\
0 & 1 & 0 & -1 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & -1 \\
0 & 0 & 0 & 1 & 0 & -1 & 0 \\
-1 & -1 & 2 & 0 & 0 & 0 & 0 \\
-1 & 0 & 1 & 0 & 1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 & 0 & 0
\end{bmatrix}
</math>

Let <math>A(a,b) = [a, b, 0, 0, 0, 0, 0]</math>, then it follows the rules:

<math display="block">\begin{array}{lcl}
[1,0,\dots] & \to^* & A(1, 2) \\
A(a, b) & \to^* & A(a-b, 4b+2) & \text{if } a > b \\
A(a, b) & \to^* & A(2a+1, b-a) & \text{if } a < b \\
A(a, b) & \to^* & \text{Halt} & \text{if } a = b
\end{array}</math>

=== BMO 6 (“Space Needle”) ===
[[File:Space Needle.webp|alt=Partial space-time diagram of Space Needle.|thumb|Partial space-time diagram of Space Needle.]]
A size 48 program was produced by hand to simulate [https://wiki.bbchallenge.org/wiki/1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD BMO 6] rules ([https://discord.com/channels/960643023006490684/1438019511155691521/1441137371046482071 Discord])

<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>

<math>\begin{bmatrix}
-1 & 0 & 0 & 1 & 1 & 0 & 0 & 0 \\
1 & -2 & 0 & 0 & -1 & 0 & 0 & 0 \\
0 & -1 & 0 & 0 & -1 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 \\
0 & 1 & 1 & -1 & 0 & 0 & -1 & 1 \\
0 & 0 & 0 & 0 & 0 & 0 & 1 & -1 \\
0 & -1 & 3 & 0 & 1 & 0 & -1 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 \\
0 & 1 & -1 & 0 & 0 & 0 & 0 & 0 \\
0 & 5 & 0 & -1 & 0 & 0 & 0 & 0 \\
1 & 0 & 1 & 0 & 0 & -1 & 0 & 0
\end{bmatrix}</math><pre>A(a, b) = B^a C^b E or B^(a-2) C^b D E

Start: A(7, 1)

A(1, b) --> halt

A(2a, b) --> A(5a+b+2, 1)

A(2a+1, b) --> A(b-1, b+c+3)</pre>

== References ==
<references />

[[Category:Functions]]

Cryptids

2026-04-09T05:38:58Z

C7X: Fix typo /* Cryptids at the Edge */

[[File:Lovecraft beaver.png|alt=A monstrous beaver in the style of HP Lovecraft with pink tentacles coming out of its mouth, 5 red spider eyes, horns on its head, elbows and tail, moss colored fur, sharp purple claws and webbed feet.|thumb|Lovecraftian Beaver fan art made by Lauren]]
'''Cryptids''' are Turing Machines whose behavior (when started on a blank tape) can be described completely by a relatively simple mathematical rule, but where that rule falls into a class of unsolved (and presumed hard) mathematical problems. This definition is somewhat subjective (What counts as a simple rule? What counts as a hard problem?). In practice, most currently known small Cryptids have [[Collatz-like]] behavior. In other words, the halting problem from blank tape of Cryptids is mathematically-hard.

If there exists a Cryptid with n states and m symbols, then BB(n, m) cannot be solved without solving this hard math problem.

The name Cryptid was proposed by Shawn Ligocki in an Oct 2023 [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html blog post] announcing the discovery of [[Bigfoot]].

== Cryptids at the Edge ==

This is a list of notable Minimal Cryptids (Cryptids in a [[:Category:BB_Domains|domain]] with no strictly smaller known Cryptid). All of these Cryptids were "discovered in the wild" rather than "constructed".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|
|-
|[[Hydra]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|
|-
|[[Bonus cryptid]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|June 2024
|<code>@mxdys</code>, shown to be a Cryptid by <code>@racheline</code>.
|Same as '''Hydra''' but starting iteration from 8 instead of 3 and with termination condition <code>O > 2E</code> instead of <code>E > 2O</code>, hence the name '''Antihydra'''.
|-
|[[Lucy's Moonlight]]
|[[BB(6)]]
|{{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}
|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|Aug 2024
|<code>mxdys</code>
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}
|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|Nov 2024
|Racheline
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|Space Needle]]
|[[BB(6)]]
|{{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}
|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|Jul 2024
|<code>mxdys</code>
|Has near-identical behavior to 16 related BB(6) holdouts. Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}}
|Jul 2024
|Racheline
|Probviously decidable. Estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
|Aug 2024
|mxdys, shown to be a Cryptid by DrDisentangle
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE|undecided}}
|April 2026
|Sheep, shown to be a Cryptid by Daniel Yuan
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB|undecided}}
|Feb 2025
|Racheline
|Probviously halting.
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Probviously non-halting.
|}
The following machines have chaotic behavior, but have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}} [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]

== Larger Cryptids ==

A more complete list of notable known Cryptids over a wider range of states and symbols. These Cryptids were all "constructed" rather than "discovered".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Announcement !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF]]
|BB(432)
|style="width:30%;word-break:break-word"|Wade's machine: https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e
|
|2025
|Wade, based on work by CatIsFluffy and O'Rear
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|-
|PA
|BB(372)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord message]
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent.
|-
|RH
|BB(744)
|style="width:30%;word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/riemann-matiyasevich-aaronson.nql
|
|2016
|Matiyasevich and O’Rear
|The machine halts if and only if [https://en.wikipedia.org/wiki/Riemann_hypothesis Riemann Hypothesis] is false.
|-
|Goldbach
|BB(25)
|style="width:30%;word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RD_1LC1RB_0RA1LC_0LQ1RE_0LF1RG_0LC1LF_0LF0LH_1LQ1LI_0RJ0LI_1RK0LJ_0RL0RS_1RL0RM_1RN1RM_0LO0LU_0LP1LO_1RH1LX_1LR1LQ_0RK0LT_1LR1RS_---1RC_1LV1LU_0LW0LJ_0RK0LW_1RY1LX_1RE1RY</pre></div></div>
|
|2016
|anonymous
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
| Erdős
| BB(5,4) and BB(15)
|style="width:30%;word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE</pre></div></div>
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA</pre></div></div>
|| [https://arxiv.org/abs/2107.12475 arxiv preprint] || Jul 2021 || [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods || The machine halts if and only if the following conjecture by Erdős is false: "For all n > 8, there is at least one 2 in the base-3 representation of 2^n."
|-
|Weak Collatz
|BB(124) and BB(43,4)
|style="width:30%;word-break:break-word"|https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt (unverified)
https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt (unverified)
|
|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The machine halts if and only if the "weak Collatz conjecture" is false. The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers.
Not independently verified, and probably easy to further optimise.
|-
|Fermat's Last Theorem
|BB(400)
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|Doctoral dissertation, ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965)
|1965
|Randels
|The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994. The machine is not explicitly given, but there is a flowchart to construct the machine, and the state count is claimed to be less than 400.
|-
| Bigfoot - compiled|| [[BB(7)]]||style="width:30%;word-break:break-word"| <code>0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB</code>|| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment] || June 2024 || <code>@Iijil1</code>|| Compilation of Bigfoot into 2 symbols, there was a previous compilation [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 with 8 states]
|-
| Hydra - compiled
|BB(9)
|style="width:30%;word-break:break-word"|<pre>
0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ
</pre>[[File:Hydra_9_states.txt]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1251572501578780782 Discord message]
|June 2024
|<code>@Iijil1</code>
|Compilation of Hydra into 2 symbols, all [https://discord.com/channels/960643023006490684/1084047886494470185/1253193750486974464 confirmed by Shawn Ligocki]. <code>@Iijil1</code> provided 24 TMs which all emulate the same behavior.
<small>[https://discord.com/channels/960643023006490684/1084047886494470185/1247560072427474955 Previous compilation had 10 states], by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|}

== Beeping Busy Beaver ==

Cryptids were actually noticed in the [[Beeping Busy Beaver]] problem before they were in the classic Busy Beaver. See [[Mother of Giants]] describing a "family" of Turing machines which "[[probviously]]" [[quasihalt]], but requires solving a Collatz-like problem in order to actually prove it. They are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-04-09T05:37:22Z

C7X: Fix year /* Larger Cryptids */

[[File:Lovecraft beaver.png|alt=A monstrous beaver in the style of HP Lovecraft with pink tentacles coming out of its mouth, 5 red spider eyes, horns on its head, elbows and tail, moss colored fur, sharp purple claws and webbed feet.|thumb|Lovecraftian Beaver fan art made by Lauren]]
'''Cryptids''' are Turing Machines whose behavior (when started on a blank tape) can be described completely by a relatively simple mathematical rule, but where that rule falls into a class of unsolved (and presumed hard) mathematical problems. This definition is somewhat subjective (What counts as a simple rule? What counts as a hard problem?). In practice, most currently known small Cryptids have [[Collatz-like]] behavior. In other words, the halting problem from blank tape of Cryptids is mathematically-hard.

If there exists a Cryptid with n states and m symbols, then BB(n, m) cannot be solved without solving this hard math problem.

The name Cryptid was proposed by Shawn Ligocki in an Oct 2023 [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html blog post] announcing the discovery of [[Bigfoot]].

== Cryptids at the Edge ==

This is a list of notable Minimal Cryptids (Cryptids in a [[:Category:BB_Domains|domain]] with no strictly smaller known Cryptid). All of these Cryptids were "discovered in the wild" rather than "constructed".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|
|-
|[[Hydra]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|
|-
|[[Bonus cryptid]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|June 2024
|<code>@mxdys</code>, shown to be a Cryptid by <code>@racheline</code>.
|Same as '''Hydra''' but starting iteration from 8 instead of 3 and with termination condition <code>O > 2E</code> instead of <code>E > 2O</code>, hence the name '''Antihydra'''.
|-
|[[Lucy's Moonlight]]
|[[BB(6)]]
|{{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}
|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|Aug 2024
|<code>mxdys</code>
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}
|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|Nov 2024
|Racheline
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|Space Needle]]
|[[BB(6)]]
|{{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}
|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|Jul 2024
|<code>mxdys</code>
|Has near-identical behavior to 16 related BB(6) holdouts. Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}}
|Jul 2024
|Racheline
|Probviously decidable. Estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
|Aug 2024
|mxdys, shown to be a Cryptid by DrDisentangle
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE|undecided}}
|April 2026
|Sheep, shown to be a Cryptid by Daniel Yuan
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB|undecided}}
|Feb 2025
|Racheline
|probvioulsy halting
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Probviously non-halting.
|}
The following machines have chaotic behavior, but have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}} [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]

== Larger Cryptids ==

A more complete list of notable known Cryptids over a wider range of states and symbols. These Cryptids were all "constructed" rather than "discovered".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Announcement !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF]]
|BB(432)
|style="width:30%;word-break:break-word"|Wade's machine: https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e
|
|2025
|Wade, based on work by CatIsFluffy and O'Rear
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|-
|PA
|BB(372)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord message]
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent.
|-
|RH
|BB(744)
|style="width:30%;word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/riemann-matiyasevich-aaronson.nql
|
|2016
|Matiyasevich and O’Rear
|The machine halts if and only if [https://en.wikipedia.org/wiki/Riemann_hypothesis Riemann Hypothesis] is false.
|-
|Goldbach
|BB(25)
|style="width:30%;word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RD_1LC1RB_0RA1LC_0LQ1RE_0LF1RG_0LC1LF_0LF0LH_1LQ1LI_0RJ0LI_1RK0LJ_0RL0RS_1RL0RM_1RN1RM_0LO0LU_0LP1LO_1RH1LX_1LR1LQ_0RK0LT_1LR1RS_---1RC_1LV1LU_0LW0LJ_0RK0LW_1RY1LX_1RE1RY</pre></div></div>
|
|2016
|anonymous
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
| Erdős
| BB(5,4) and BB(15)
|style="width:30%;word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE</pre></div></div>
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA</pre></div></div>
|| [https://arxiv.org/abs/2107.12475 arxiv preprint] || Jul 2021 || [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods || The machine halts if and only if the following conjecture by Erdős is false: "For all n > 8, there is at least one 2 in the base-3 representation of 2^n."
|-
|Weak Collatz
|BB(124) and BB(43,4)
|style="width:30%;word-break:break-word"|https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt (unverified)
https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt (unverified)
|
|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The machine halts if and only if the "weak Collatz conjecture" is false. The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers.
Not independently verified, and probably easy to further optimise.
|-
|Fermat's Last Theorem
|BB(400)
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|Doctoral dissertation, ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965)
|1965
|Randels
|The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994. The machine is not explicitly given, but there is a flowchart to construct the machine, and the state count is claimed to be less than 400.
|-
| Bigfoot - compiled|| [[BB(7)]]||style="width:30%;word-break:break-word"| <code>0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB</code>|| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment] || June 2024 || <code>@Iijil1</code>|| Compilation of Bigfoot into 2 symbols, there was a previous compilation [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 with 8 states]
|-
| Hydra - compiled
|BB(9)
|style="width:30%;word-break:break-word"|<pre>
0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ
</pre>[[File:Hydra_9_states.txt]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1251572501578780782 Discord message]
|June 2024
|<code>@Iijil1</code>
|Compilation of Hydra into 2 symbols, all [https://discord.com/channels/960643023006490684/1084047886494470185/1253193750486974464 confirmed by Shawn Ligocki]. <code>@Iijil1</code> provided 24 TMs which all emulate the same behavior.
<small>[https://discord.com/channels/960643023006490684/1084047886494470185/1247560072427474955 Previous compilation had 10 states], by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|}

== Beeping Busy Beaver ==

Cryptids were actually noticed in the [[Beeping Busy Beaver]] problem before they were in the classic Busy Beaver. See [[Mother of Giants]] describing a "family" of Turing machines which "[[probviously]]" [[quasihalt]], but requires solving a Collatz-like problem in order to actually prove it. They are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-04-09T05:37:07Z

C7X: Fix page numbers /* Larger Cryptids */

[[File:Lovecraft beaver.png|alt=A monstrous beaver in the style of HP Lovecraft with pink tentacles coming out of its mouth, 5 red spider eyes, horns on its head, elbows and tail, moss colored fur, sharp purple claws and webbed feet.|thumb|Lovecraftian Beaver fan art made by Lauren]]
'''Cryptids''' are Turing Machines whose behavior (when started on a blank tape) can be described completely by a relatively simple mathematical rule, but where that rule falls into a class of unsolved (and presumed hard) mathematical problems. This definition is somewhat subjective (What counts as a simple rule? What counts as a hard problem?). In practice, most currently known small Cryptids have [[Collatz-like]] behavior. In other words, the halting problem from blank tape of Cryptids is mathematically-hard.

If there exists a Cryptid with n states and m symbols, then BB(n, m) cannot be solved without solving this hard math problem.

The name Cryptid was proposed by Shawn Ligocki in an Oct 2023 [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html blog post] announcing the discovery of [[Bigfoot]].

== Cryptids at the Edge ==

This is a list of notable Minimal Cryptids (Cryptids in a [[:Category:BB_Domains|domain]] with no strictly smaller known Cryptid). All of these Cryptids were "discovered in the wild" rather than "constructed".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|
|-
|[[Hydra]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|
|-
|[[Bonus cryptid]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|June 2024
|<code>@mxdys</code>, shown to be a Cryptid by <code>@racheline</code>.
|Same as '''Hydra''' but starting iteration from 8 instead of 3 and with termination condition <code>O > 2E</code> instead of <code>E > 2O</code>, hence the name '''Antihydra'''.
|-
|[[Lucy's Moonlight]]
|[[BB(6)]]
|{{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}
|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|Aug 2024
|<code>mxdys</code>
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}
|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|Nov 2024
|Racheline
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|Space Needle]]
|[[BB(6)]]
|{{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}
|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|Jul 2024
|<code>mxdys</code>
|Has near-identical behavior to 16 related BB(6) holdouts. Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}}
|Jul 2024
|Racheline
|Probviously decidable. Estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
|Aug 2024
|mxdys, shown to be a Cryptid by DrDisentangle
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE|undecided}}
|April 2026
|Sheep, shown to be a Cryptid by Daniel Yuan
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB|undecided}}
|Feb 2025
|Racheline
|probvioulsy halting
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Probviously non-halting.
|}
The following machines have chaotic behavior, but have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}} [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]

== Larger Cryptids ==

A more complete list of notable known Cryptids over a wider range of states and symbols. These Cryptids were all "constructed" rather than "discovered".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Announcement !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF]]
|BB(432)
|style="width:30%;word-break:break-word"|Wade's machine: https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e
|
|2025
|Wade, based on work by CatIsFluffy and O'Rear
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|-
|PA
|BB(372)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord message]
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent.
|-
|RH
|BB(744)
|style="width:30%;word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/riemann-matiyasevich-aaronson.nql
|
|2016
|Matiyasevich and O’Rear
|The machine halts if and only if [https://en.wikipedia.org/wiki/Riemann_hypothesis Riemann Hypothesis] is false.
|-
|Goldbach
|BB(25)
|style="width:30%;word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RD_1LC1RB_0RA1LC_0LQ1RE_0LF1RG_0LC1LF_0LF0LH_1LQ1LI_0RJ0LI_1RK0LJ_0RL0RS_1RL0RM_1RN1RM_0LO0LU_0LP1LO_1RH1LX_1LR1LQ_0RK0LT_1LR1RS_---1RC_1LV1LU_0LW0LJ_0RK0LW_1RY1LX_1RE1RY</pre></div></div>
|
|2016
|anonymous
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
| Erdős
| BB(5,4) and BB(15)
|style="width:30%;word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE</pre></div></div>
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA</pre></div></div>
|| [https://arxiv.org/abs/2107.12475 arxiv preprint] || Jul 2021 || [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods || The machine halts if and only if the following conjecture by Erdős is false: "For all n > 8, there is at least one 2 in the base-3 representation of 2^n."
|-
|Weak Collatz
|BB(124) and BB(43,4)
|style="width:30%;word-break:break-word"|https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt (unverified)
https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt (unverified)
|
|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The machine halts if and only if the "weak Collatz conjecture" is false. The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers.
Not independently verified, and probably easy to further optimise.
|-
|Fermat's Last Theorem
|BB(400)
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|Doctoral dissertation, ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965)
|Jul 2021
|Randels
|The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994. The machine is not explicitly given, but there is a flowchart to construct the machine, and the state count is claimed to be less than 400.
|-
| Bigfoot - compiled|| [[BB(7)]]||style="width:30%;word-break:break-word"| <code>0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB</code>|| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment] || June 2024 || <code>@Iijil1</code>|| Compilation of Bigfoot into 2 symbols, there was a previous compilation [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 with 8 states]
|-
| Hydra - compiled
|BB(9)
|style="width:30%;word-break:break-word"|<pre>
0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ
</pre>[[File:Hydra_9_states.txt]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1251572501578780782 Discord message]
|June 2024
|<code>@Iijil1</code>
|Compilation of Hydra into 2 symbols, all [https://discord.com/channels/960643023006490684/1084047886494470185/1253193750486974464 confirmed by Shawn Ligocki]. <code>@Iijil1</code> provided 24 TMs which all emulate the same behavior.
<small>[https://discord.com/channels/960643023006490684/1084047886494470185/1247560072427474955 Previous compilation had 10 states], by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|}

== Beeping Busy Beaver ==

Cryptids were actually noticed in the [[Beeping Busy Beaver]] problem before they were in the classic Busy Beaver. See [[Mother of Giants]] describing a "family" of Turing machines which "[[probviously]]" [[quasihalt]], but requires solving a Collatz-like problem in order to actually prove it. They are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-04-09T05:35:26Z

C7X: /* Larger Cryptids */

[[File:Lovecraft beaver.png|alt=A monstrous beaver in the style of HP Lovecraft with pink tentacles coming out of its mouth, 5 red spider eyes, horns on its head, elbows and tail, moss colored fur, sharp purple claws and webbed feet.|thumb|Lovecraftian Beaver fan art made by Lauren]]
'''Cryptids''' are Turing Machines whose behavior (when started on a blank tape) can be described completely by a relatively simple mathematical rule, but where that rule falls into a class of unsolved (and presumed hard) mathematical problems. This definition is somewhat subjective (What counts as a simple rule? What counts as a hard problem?). In practice, most currently known small Cryptids have [[Collatz-like]] behavior. In other words, the halting problem from blank tape of Cryptids is mathematically-hard.

If there exists a Cryptid with n states and m symbols, then BB(n, m) cannot be solved without solving this hard math problem.

The name Cryptid was proposed by Shawn Ligocki in an Oct 2023 [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html blog post] announcing the discovery of [[Bigfoot]].

== Cryptids at the Edge ==

This is a list of notable Minimal Cryptids (Cryptids in a [[:Category:BB_Domains|domain]] with no strictly smaller known Cryptid). All of these Cryptids were "discovered in the wild" rather than "constructed".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|
|-
|[[Hydra]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|
|-
|[[Bonus cryptid]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|June 2024
|<code>@mxdys</code>, shown to be a Cryptid by <code>@racheline</code>.
|Same as '''Hydra''' but starting iteration from 8 instead of 3 and with termination condition <code>O > 2E</code> instead of <code>E > 2O</code>, hence the name '''Antihydra'''.
|-
|[[Lucy's Moonlight]]
|[[BB(6)]]
|{{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}
|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|Aug 2024
|<code>mxdys</code>
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}
|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|Nov 2024
|Racheline
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|Space Needle]]
|[[BB(6)]]
|{{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}
|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|Jul 2024
|<code>mxdys</code>
|Has near-identical behavior to 16 related BB(6) holdouts. Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}}
|Jul 2024
|Racheline
|Probviously decidable. Estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
|Aug 2024
|mxdys, shown to be a Cryptid by DrDisentangle
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE|undecided}}
|April 2026
|Sheep, shown to be a Cryptid by Daniel Yuan
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB|undecided}}
|Feb 2025
|Racheline
|probvioulsy halting
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Probviously non-halting.
|}
The following machines have chaotic behavior, but have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}} [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]

== Larger Cryptids ==

A more complete list of notable known Cryptids over a wider range of states and symbols. These Cryptids were all "constructed" rather than "discovered".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Announcement !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF]]
|BB(432)
|style="width:30%;word-break:break-word"|Wade's machine: https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e
|
|2025
|Wade, based on work by CatIsFluffy and O'Rear
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|-
|PA
|BB(372)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord message]
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent.
|-
|RH
|BB(744)
|style="width:30%;word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/riemann-matiyasevich-aaronson.nql
|
|2016
|Matiyasevich and O’Rear
|The machine halts if and only if [https://en.wikipedia.org/wiki/Riemann_hypothesis Riemann Hypothesis] is false.
|-
|Goldbach
|BB(25)
|style="width:30%;word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RD_1LC1RB_0RA1LC_0LQ1RE_0LF1RG_0LC1LF_0LF0LH_1LQ1LI_0RJ0LI_1RK0LJ_0RL0RS_1RL0RM_1RN1RM_0LO0LU_0LP1LO_1RH1LX_1LR1LQ_0RK0LT_1LR1RS_---1RC_1LV1LU_0LW0LJ_0RK0LW_1RY1LX_1RE1RY</pre></div></div>
|
|2016
|anonymous
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
| Erdős
| BB(5,4) and BB(15)
|style="width:30%;word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE</pre></div></div>
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA</pre></div></div>
|| [https://arxiv.org/abs/2107.12475 arxiv preprint] || Jul 2021 || [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods || The machine halts if and only if the following conjecture by Erdős is false: "For all n > 8, there is at least one 2 in the base-3 representation of 2^n."
|-
|Weak Collatz
|BB(124) and BB(43,4)
|style="width:30%;word-break:break-word"|https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt (unverified)
https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt (unverified)
|
|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The machine halts if and only if the "weak Collatz conjecture" is false. The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers.
Not independently verified, and probably easy to further optimise.
|-
|Fermat's Last Theorem
|BB(400)
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), p. 81
|Doctoral dissertation, ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965)
|Jul 2021
|Randels
|The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994. The machine is not explicitly given, but there is a flowchart to construct the machine, and the state count is claimed to be less than 400.
|-
| Bigfoot - compiled|| [[BB(7)]]||style="width:30%;word-break:break-word"| <code>0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB</code>|| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment] || June 2024 || <code>@Iijil1</code>|| Compilation of Bigfoot into 2 symbols, there was a previous compilation [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 with 8 states]
|-
| Hydra - compiled
|BB(9)
|style="width:30%;word-break:break-word"|<pre>
0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ
</pre>[[File:Hydra_9_states.txt]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1251572501578780782 Discord message]
|June 2024
|<code>@Iijil1</code>
|Compilation of Hydra into 2 symbols, all [https://discord.com/channels/960643023006490684/1084047886494470185/1253193750486974464 confirmed by Shawn Ligocki]. <code>@Iijil1</code> provided 24 TMs which all emulate the same behavior.
<small>[https://discord.com/channels/960643023006490684/1084047886494470185/1247560072427474955 Previous compilation had 10 states], by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|}

== Beeping Busy Beaver ==

Cryptids were actually noticed in the [[Beeping Busy Beaver]] problem before they were in the classic Busy Beaver. See [[Mother of Giants]] describing a "family" of Turing machines which "[[probviously]]" [[quasihalt]], but requires solving a Collatz-like problem in order to actually prove it. They are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-04-09T05:34:40Z

C7X: Add period /* Larger Cryptids */

[[File:Lovecraft beaver.png|alt=A monstrous beaver in the style of HP Lovecraft with pink tentacles coming out of its mouth, 5 red spider eyes, horns on its head, elbows and tail, moss colored fur, sharp purple claws and webbed feet.|thumb|Lovecraftian Beaver fan art made by Lauren]]
'''Cryptids''' are Turing Machines whose behavior (when started on a blank tape) can be described completely by a relatively simple mathematical rule, but where that rule falls into a class of unsolved (and presumed hard) mathematical problems. This definition is somewhat subjective (What counts as a simple rule? What counts as a hard problem?). In practice, most currently known small Cryptids have [[Collatz-like]] behavior. In other words, the halting problem from blank tape of Cryptids is mathematically-hard.

If there exists a Cryptid with n states and m symbols, then BB(n, m) cannot be solved without solving this hard math problem.

The name Cryptid was proposed by Shawn Ligocki in an Oct 2023 [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html blog post] announcing the discovery of [[Bigfoot]].

== Cryptids at the Edge ==

This is a list of notable Minimal Cryptids (Cryptids in a [[:Category:BB_Domains|domain]] with no strictly smaller known Cryptid). All of these Cryptids were "discovered in the wild" rather than "constructed".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|
|-
|[[Hydra]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|
|-
|[[Bonus cryptid]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|June 2024
|<code>@mxdys</code>, shown to be a Cryptid by <code>@racheline</code>.
|Same as '''Hydra''' but starting iteration from 8 instead of 3 and with termination condition <code>O > 2E</code> instead of <code>E > 2O</code>, hence the name '''Antihydra'''.
|-
|[[Lucy's Moonlight]]
|[[BB(6)]]
|{{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}
|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|Aug 2024
|<code>mxdys</code>
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}
|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|Nov 2024
|Racheline
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|Space Needle]]
|[[BB(6)]]
|{{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}
|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|Jul 2024
|<code>mxdys</code>
|Has near-identical behavior to 16 related BB(6) holdouts. Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}}
|Jul 2024
|Racheline
|Probviously decidable. Estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
|Aug 2024
|mxdys, shown to be a Cryptid by DrDisentangle
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE|undecided}}
|April 2026
|Sheep, shown to be a Cryptid by Daniel Yuan
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB|undecided}}
|Feb 2025
|Racheline
|probvioulsy halting
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Probviously non-halting.
|}
The following machines have chaotic behavior, but have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}} [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]

== Larger Cryptids ==

A more complete list of notable known Cryptids over a wider range of states and symbols. These Cryptids were all "constructed" rather than "discovered".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Announcement !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF]]
|BB(432)
|style="width:30%;word-break:break-word"|Wade's machine: https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e
|
|2025
|Wade, based on work by CatIsFluffy and O'Rear
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|-
|PA
|BB(372)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord message]
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent.
|-
|RH
|BB(744)
|style="width:30%;word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/riemann-matiyasevich-aaronson.nql
|
|2016
|Matiyasevich and O’Rear
|The machine halts if and only if [https://en.wikipedia.org/wiki/Riemann_hypothesis Riemann Hypothesis] is false.
|-
|Goldbach
|BB(25)
|style="width:30%;word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RD_1LC1RB_0RA1LC_0LQ1RE_0LF1RG_0LC1LF_0LF0LH_1LQ1LI_0RJ0LI_1RK0LJ_0RL0RS_1RL0RM_1RN1RM_0LO0LU_0LP1LO_1RH1LX_1LR1LQ_0RK0LT_1LR1RS_---1RC_1LV1LU_0LW0LJ_0RK0LW_1RY1LX_1RE1RY</pre></div></div>
|
|2016
|anonymous
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
| Erdős
| BB(5,4) and BB(15)
|style="width:30%;word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE</pre></div></div>
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA</pre></div></div>
|| [https://arxiv.org/abs/2107.12475 arxiv preprint] || Jul 2021 || [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods || The machine halts if and only if the following conjecture by Erdős is false: "For all n > 8, there is at least one 2 in the base-3 representation of 2^n."
|-
|Weak Collatz
|BB(124) and BB(43,4)
|style="width:30%;word-break:break-word"|https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt (unverified)
https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt (unverified)
|
|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The machine halts if and only if the "weak Collatz conjecture" is false. The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers.
Not independently verified, and probably easy to further optimise.
|-
|Fermat's Last Theorem
|BB(400)
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), p. 81
|Doctoral dissertation, ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965)
|Jul 2021
|Randels
|The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994. A flowchart to construct the machine is given, the state count is claimed to be less than 400.
|-
| Bigfoot - compiled|| [[BB(7)]]||style="width:30%;word-break:break-word"| <code>0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB</code>|| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment] || June 2024 || <code>@Iijil1</code>|| Compilation of Bigfoot into 2 symbols, there was a previous compilation [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 with 8 states]
|-
| Hydra - compiled
|BB(9)
|style="width:30%;word-break:break-word"|<pre>
0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ
</pre>[[File:Hydra_9_states.txt]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1251572501578780782 Discord message]
|June 2024
|<code>@Iijil1</code>
|Compilation of Hydra into 2 symbols, all [https://discord.com/channels/960643023006490684/1084047886494470185/1253193750486974464 confirmed by Shawn Ligocki]. <code>@Iijil1</code> provided 24 TMs which all emulate the same behavior.
<small>[https://discord.com/channels/960643023006490684/1084047886494470185/1247560072427474955 Previous compilation had 10 states], by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|}

== Beeping Busy Beaver ==

Cryptids were actually noticed in the [[Beeping Busy Beaver]] problem before they were in the classic Busy Beaver. See [[Mother of Giants]] describing a "family" of Turing machines which "[[probviously]]" [[quasihalt]], but requires solving a Collatz-like problem in order to actually prove it. They are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-04-09T05:34:17Z

C7X: /* Larger Cryptids */

[[File:Lovecraft beaver.png|alt=A monstrous beaver in the style of HP Lovecraft with pink tentacles coming out of its mouth, 5 red spider eyes, horns on its head, elbows and tail, moss colored fur, sharp purple claws and webbed feet.|thumb|Lovecraftian Beaver fan art made by Lauren]]
'''Cryptids''' are Turing Machines whose behavior (when started on a blank tape) can be described completely by a relatively simple mathematical rule, but where that rule falls into a class of unsolved (and presumed hard) mathematical problems. This definition is somewhat subjective (What counts as a simple rule? What counts as a hard problem?). In practice, most currently known small Cryptids have [[Collatz-like]] behavior. In other words, the halting problem from blank tape of Cryptids is mathematically-hard.

If there exists a Cryptid with n states and m symbols, then BB(n, m) cannot be solved without solving this hard math problem.

The name Cryptid was proposed by Shawn Ligocki in an Oct 2023 [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html blog post] announcing the discovery of [[Bigfoot]].

== Cryptids at the Edge ==

This is a list of notable Minimal Cryptids (Cryptids in a [[:Category:BB_Domains|domain]] with no strictly smaller known Cryptid). All of these Cryptids were "discovered in the wild" rather than "constructed".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|
|-
|[[Hydra]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|
|-
|[[Bonus cryptid]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|June 2024
|<code>@mxdys</code>, shown to be a Cryptid by <code>@racheline</code>.
|Same as '''Hydra''' but starting iteration from 8 instead of 3 and with termination condition <code>O > 2E</code> instead of <code>E > 2O</code>, hence the name '''Antihydra'''.
|-
|[[Lucy's Moonlight]]
|[[BB(6)]]
|{{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}
|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|Aug 2024
|<code>mxdys</code>
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}
|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|Nov 2024
|Racheline
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|Space Needle]]
|[[BB(6)]]
|{{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}
|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|Jul 2024
|<code>mxdys</code>
|Has near-identical behavior to 16 related BB(6) holdouts. Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}}
|Jul 2024
|Racheline
|Probviously decidable. Estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
|Aug 2024
|mxdys, shown to be a Cryptid by DrDisentangle
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE|undecided}}
|April 2026
|Sheep, shown to be a Cryptid by Daniel Yuan
|Similar to Space Needle, probviously non-halting
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB|undecided}}
|Feb 2025
|Racheline
|probvioulsy halting
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Probviously non-halting.
|}
The following machines have chaotic behavior, but have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}} [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]

== Larger Cryptids ==

A more complete list of notable known Cryptids over a wider range of states and symbols. These Cryptids were all "constructed" rather than "discovered".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Announcement !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF]]
|BB(432)
|style="width:30%;word-break:break-word"|Wade's machine: https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e
|
|2025
|Wade, based on work by CatIsFluffy and O'Rear
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|-
|PA
|BB(372)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord message]
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent.
|-
|RH
|BB(744)
|style="width:30%;word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/riemann-matiyasevich-aaronson.nql
|
|2016
|Matiyasevich and O’Rear
|The machine halts if and only if [https://en.wikipedia.org/wiki/Riemann_hypothesis Riemann Hypothesis] is false.
|-
|Goldbach
|BB(25)
|style="width:30%;word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RD_1LC1RB_0RA1LC_0LQ1RE_0LF1RG_0LC1LF_0LF0LH_1LQ1LI_0RJ0LI_1RK0LJ_0RL0RS_1RL0RM_1RN1RM_0LO0LU_0LP1LO_1RH1LX_1LR1LQ_0RK0LT_1LR1RS_---1RC_1LV1LU_0LW0LJ_0RK0LW_1RY1LX_1RE1RY</pre></div></div>
|
|2016
|anonymous
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
| Erdős
| BB(5,4) and BB(15)
|style="width:30%;word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE</pre></div></div>
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA</pre></div></div>
|| [https://arxiv.org/abs/2107.12475 arxiv preprint] || Jul 2021 || [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods || The machine halts if and only if the following conjecture by Erdős is false: "For all n > 8, there is at least one 2 in the base-3 representation of 2^n"
|-
|Weak Collatz
|BB(124) and BB(43,4)
|style="width:30%;word-break:break-word"|https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt (unverified)
https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt (unverified)
|
|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The machine halts if and only if the "weak Collatz conjecture" is false. The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers.
Not independently verified, and probably easy to further optimise.
|-
|Fermat's Last Theorem
|BB(400)
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), p. 81
|Doctoral dissertation, ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965)
|Jul 2021
|Randels
|The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994. A flowchart to construct the machine is given, the state count is claimed to be less than 400.
|-
| Bigfoot - compiled|| [[BB(7)]]||style="width:30%;word-break:break-word"| <code>0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB</code>|| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment] || June 2024 || <code>@Iijil1</code>|| Compilation of Bigfoot into 2 symbols, there was a previous compilation [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 with 8 states]
|-
| Hydra - compiled
|BB(9)
|style="width:30%;word-break:break-word"|<pre>
0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ
</pre>[[File:Hydra_9_states.txt]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1251572501578780782 Discord message]
|June 2024
|<code>@Iijil1</code>
|Compilation of Hydra into 2 symbols, all [https://discord.com/channels/960643023006490684/1084047886494470185/1253193750486974464 confirmed by Shawn Ligocki]. <code>@Iijil1</code> provided 24 TMs which all emulate the same behavior.
<small>[https://discord.com/channels/960643023006490684/1084047886494470185/1247560072427474955 Previous compilation had 10 states], by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|}

== Beeping Busy Beaver ==

Cryptids were actually noticed in the [[Beeping Busy Beaver]] problem before they were in the classic Busy Beaver. See [[Mother of Giants]] describing a "family" of Turing machines which "[[probviously]]" [[quasihalt]], but requires solving a Collatz-like problem in order to actually prove it. They are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.
[[Category:Zoology]]
[[Category:Cryptids]]

Graham's number

2026-03-14T18:36:55Z

C7X: Correction

'''Graham's number''' (<math>g_{64}</math> or <math>G</math>) is a famously huge number which Martin Gardner claimed was the "largest number ever used in a serious mathematical proof" in 1977. Since it is one of the most famous large numbers, it has become a bit of a yardstick for measuring "hugeness". In the specific context of the Busy Beaver game, we can ask, what is the smallest <math>n</math> such that <math>BB(n) > g_{64}</math>. There is an active search for the smallest TM that runs for over Graham's number steps.

== Definition ==
See [https://en.wikipedia.org/wiki/Graham%27s_number Wikipedia article] for more detail
<math display="block">
\begin{array}{l}
g_0 & = & 4 \\
g_n & = & 3 \uparrow^{g_{n-1}} 3 & \text{if } n \ge 1 \\
\end{array}
</math>

Graham's number is <math>g_{64}</math>.

Using the [[fast-growing hierarchy]], <math>g_{64} < f_{\omega + 1}(64)</math>.

Let <math>N_G</math> be the smallest integer such that <math>S(N_G) > g_{64}</math>.

== Bounds ==
The current known bounds for <math>N_G</math> are:
<math display="block">
6 \le N_G \le 13
</math>

The lower bound comes from the proof that [[BB(5)]] = 47,176,870 and the upper bound from a specific 13-state TM found by 50_ft_lock in March 2026 which runs for <math> > f_{\omega + 1}(2\,047) > g_{64} </math> steps.

=== History of Graham-beating TMs ===
There is no one authoritative source on the history of TMs beating Graham's number. Most were posted in personal blog posts, Googology pages/comments or on the bbchallenge Discord. They are often unverified and sometimes undocumented. This list is based upon historical accounts listed on [https://googology.fandom.com/wiki/Graham%27s_number#Comparison_with_busy_beaver_function Googologogy wiki], [https://cs.stackexchange.com/questions/69469/what-is-the-smallest-n-such-that-bbn-grahams-number/69476#69476 a 2017 CS Stack Exchange answer], [https://www.sligocki.com/2010/07/04/beating-grahams-number.html#results-from-the-future a 2022 history synthesis by Shawn Ligocki] and [https://bbchallenge.org/~pascal.michel/ha#topdown summary by Pascal Michel]. In general, these results are self-reported and we do not know of independent verification for most of them. Most of these were actually proven as bounds for the <math>\Sigma</math> function, but since <math>S(n) \ge \Sigma(n)</math> they apply to this formulation as well.
{| class="wikitable"
|+History of Graham-beating TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|91
|9 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|72
|13 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|64
|19 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/64-state-turing-machine-whose-output.html Blog Post] [https://web.archive.org/web/20130509222047/https://sites.google.com/site/res0001/surpassing-graham-s-number Summary]
|
|-
|25
|31 Mar 2013
|"Deedlit11"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines#A_New_Record!_Beating_Graham's_number_with_a_2-symbol_TM Googology Post]
|
|-
|24
|27 September 2013
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines?commentId=4400000000000011880 Googology Comment]
|
|-
|23
|7 October 2013
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines?commentId=4400000000000011880&replyId=4400000000000036301 Googology Comment]
|
|-
|22
|5 August 2014
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/NEWS!_I_found_a_22-state_machine_that_beats_G! Googology Post]
|
|-
|18
|24 Jul 2016
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham's_Number!?useskin=oasis Googology Post]
|
|-
|16
|26 Mar 2021
|Daniel Nagaj
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham%27s_Number!?commentId=4400000000000019187&replyId=4400000000000101283 Googology Comment] [http://morphett.info/turing/turing.html?197640ce0f380f8a6b0a4cdd138156a0 TM Definition]
|[https://www.sligocki.com/2022/07/11/bb-16-graham.html Analysis by Shawn Ligocki in 2022]
|-
|14
|17 Aug 2024
|[[User:Racheline|Racheline]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Discord Message]
|
|-
|13
|13 March 2026
|50_ft_lock
|[https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 Discord Message]
|}

Graham's number

2026-03-14T18:36:41Z

C7X: Update /* Bounds */

'''Graham's number''' (<math>g_{64}</math> or <math>G</math>) is a famously huge number which Martin Gardner claimed was the "largest number ever used in a serious mathematical proof" in 1977. Since it is one of the most famous large numbers, it has become a bit of a yardstick for measuring "hugeness". In the specific context of the Busy Beaver game, we can ask, what is the smallest <math>n</math> such that <math>BB(n) > g_{64}</math>. There is an active search for the smallest TM that runs for over Graham's number steps.

== Definition ==
See [https://en.wikipedia.org/wiki/Graham%27s_number Wikipedia article] for more detail
<math display="block">
\begin{array}{l}
g_0 & = & 4 \\
g_n & = & 3 \uparrow^{g_{n-1}} 3 & \text{if } n \ge 1 \\
\end{array}
</math>

Graham's number is <math>g_{64}</math>.

Using the [[fast-growing hierarchy]], <math>g_{64} < f_{\omega + 1}(64)</math>.

Let <math>N_G</math> be the smallest integer such that <math>S(N_G) > g_{64}</math>.

== Bounds ==
The current known bounds for <math>N_G</math> are:
<math display="block">
6 \le N_G \le 13
</math>

The lower bound comes from the proof that [[BB(5)]] = 47,176,870 and the upper bound from a specific 13-state TM found by 50_ft_lock in August 2024 which runs for <math> > f_{\omega + 1}(2\,047) > g_{64} </math> steps.

=== History of Graham-beating TMs ===
There is no one authoritative source on the history of TMs beating Graham's number. Most were posted in personal blog posts, Googology pages/comments or on the bbchallenge Discord. They are often unverified and sometimes undocumented. This list is based upon historical accounts listed on [https://googology.fandom.com/wiki/Graham%27s_number#Comparison_with_busy_beaver_function Googologogy wiki], [https://cs.stackexchange.com/questions/69469/what-is-the-smallest-n-such-that-bbn-grahams-number/69476#69476 a 2017 CS Stack Exchange answer], [https://www.sligocki.com/2010/07/04/beating-grahams-number.html#results-from-the-future a 2022 history synthesis by Shawn Ligocki] and [https://bbchallenge.org/~pascal.michel/ha#topdown summary by Pascal Michel]. In general, these results are self-reported and we do not know of independent verification for most of them. Most of these were actually proven as bounds for the <math>\Sigma</math> function, but since <math>S(n) \ge \Sigma(n)</math> they apply to this formulation as well.
{| class="wikitable"
|+History of Graham-beating TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|91
|9 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|72
|13 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|64
|19 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/64-state-turing-machine-whose-output.html Blog Post] [https://web.archive.org/web/20130509222047/https://sites.google.com/site/res0001/surpassing-graham-s-number Summary]
|
|-
|25
|31 Mar 2013
|"Deedlit11"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines#A_New_Record!_Beating_Graham's_number_with_a_2-symbol_TM Googology Post]
|
|-
|24
|27 September 2013
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines?commentId=4400000000000011880 Googology Comment]
|
|-
|23
|7 October 2013
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines?commentId=4400000000000011880&replyId=4400000000000036301 Googology Comment]
|
|-
|22
|5 August 2014
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/NEWS!_I_found_a_22-state_machine_that_beats_G! Googology Post]
|
|-
|18
|24 Jul 2016
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham's_Number!?useskin=oasis Googology Post]
|
|-
|16
|26 Mar 2021
|Daniel Nagaj
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham%27s_Number!?commentId=4400000000000019187&replyId=4400000000000101283 Googology Comment] [http://morphett.info/turing/turing.html?197640ce0f380f8a6b0a4cdd138156a0 TM Definition]
|[https://www.sligocki.com/2022/07/11/bb-16-graham.html Analysis by Shawn Ligocki in 2022]
|-
|14
|17 Aug 2024
|[[User:Racheline|Racheline]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Discord Message]
|
|-
|13
|13 March 2026
|50_ft_lock
|[https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 Discord Message]
|}

Graham's number

2026-03-14T18:36:08Z

C7X: /* History of Graham-beating TMs */

'''Graham's number''' (<math>g_{64}</math> or <math>G</math>) is a famously huge number which Martin Gardner claimed was the "largest number ever used in a serious mathematical proof" in 1977. Since it is one of the most famous large numbers, it has become a bit of a yardstick for measuring "hugeness". In the specific context of the Busy Beaver game, we can ask, what is the smallest <math>n</math> such that <math>BB(n) > g_{64}</math>. There is an active search for the smallest TM that runs for over Graham's number steps.

== Definition ==
See [https://en.wikipedia.org/wiki/Graham%27s_number Wikipedia article] for more detail
<math display="block">
\begin{array}{l}
g_0 & = & 4 \\
g_n & = & 3 \uparrow^{g_{n-1}} 3 & \text{if } n \ge 1 \\
\end{array}
</math>

Graham's number is <math>g_{64}</math>.

Using the [[fast-growing hierarchy]], <math>g_{64} < f_{\omega + 1}(64)</math>.

Let <math>N_G</math> be the smallest integer such that <math>S(N_G) > g_{64}</math>.

== Bounds ==
The current known bounds for <math>N_G</math> are:
<math display="block">
6 \le N_G \le 14
</math>

The lower bound comes from the proof that [[BB(5)]] = 47,176,870 and the upper bound from a specific 14-state TM found Racheline in August 2024 which runs for <math> > f_{\omega + 1}(65\,536) > g_{64} </math> steps.

=== History of Graham-beating TMs ===
There is no one authoritative source on the history of TMs beating Graham's number. Most were posted in personal blog posts, Googology pages/comments or on the bbchallenge Discord. They are often unverified and sometimes undocumented. This list is based upon historical accounts listed on [https://googology.fandom.com/wiki/Graham%27s_number#Comparison_with_busy_beaver_function Googologogy wiki], [https://cs.stackexchange.com/questions/69469/what-is-the-smallest-n-such-that-bbn-grahams-number/69476#69476 a 2017 CS Stack Exchange answer], [https://www.sligocki.com/2010/07/04/beating-grahams-number.html#results-from-the-future a 2022 history synthesis by Shawn Ligocki] and [https://bbchallenge.org/~pascal.michel/ha#topdown summary by Pascal Michel]. In general, these results are self-reported and we do not know of independent verification for most of them. Most of these were actually proven as bounds for the <math>\Sigma</math> function, but since <math>S(n) \ge \Sigma(n)</math> they apply to this formulation as well.
{| class="wikitable"
|+History of Graham-beating TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|91
|9 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|72
|13 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|64
|19 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/64-state-turing-machine-whose-output.html Blog Post] [https://web.archive.org/web/20130509222047/https://sites.google.com/site/res0001/surpassing-graham-s-number Summary]
|
|-
|25
|31 Mar 2013
|"Deedlit11"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines#A_New_Record!_Beating_Graham's_number_with_a_2-symbol_TM Googology Post]
|
|-
|24
|27 September 2013
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines?commentId=4400000000000011880 Googology Comment]
|
|-
|23
|7 October 2013
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines?commentId=4400000000000011880&replyId=4400000000000036301 Googology Comment]
|
|-
|22
|5 August 2014
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/NEWS!_I_found_a_22-state_machine_that_beats_G! Googology Post]
|
|-
|18
|24 Jul 2016
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham's_Number!?useskin=oasis Googology Post]
|
|-
|16
|26 Mar 2021
|Daniel Nagaj
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham%27s_Number!?commentId=4400000000000019187&replyId=4400000000000101283 Googology Comment] [http://morphett.info/turing/turing.html?197640ce0f380f8a6b0a4cdd138156a0 TM Definition]
|[https://www.sligocki.com/2022/07/11/bb-16-graham.html Analysis by Shawn Ligocki in 2022]
|-
|14
|17 Aug 2024
|[[User:Racheline|Racheline]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Discord Message]
|
|-
|13
|13 March 2026
|50_ft_lock
|[https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 Discord Message]
|}

Champions

2026-03-14T18:35:26Z

C7X: Move it up /* 2-Symbol TMs */

Busy Beaver '''Champions''' are the current record holding [[Turing machine|Turing machines]] which maximize a [[Busy Beaver function]]. In this article we focus specifically on the longest running TMs. Some have been proven to be the longest running of all (and so are the ultimate champion) while others are only current champions and may be usurped in the future. For smaller domains, Pascal Michel's website is the canonical source for [https://bbchallenge.org/~pascal.michel/bbc Busy Beaver champions] and the [https://bbchallenge.org/~pascal.michel/ha History of Previous Champions]. 1-state domains are omitted as [[BB(1,m)]] = 1 for m > 1.

== Trivial Champions ==
[[BB(n,1)]] = n

[[BB(1,m)]] = 1

== 2-Symbol TMs ==
Rows are blank if no champion has been found which surpasses a smaller size problem. Also take note that the <math>f_{x}(n)</math> used in the lower bounds represent the [[Fast-Growing Hierarchy]] while <math>\uparrow</math> represents [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]]. Note that most champions above 6 states are self-reported and have not been independently verified.

{| class="wikitable"
|+
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|[[BB(2)]]
|<math>6</math>
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
|[[Tibor Radó]]
|Direct Simulation
|-
|[[BB(3)]]
|<math>21</math>
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|Proven by [[Shen Lin]]
|Direct Simulation
|-
|[[BB(4)]]
|<math>107</math>
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|Allen Brady
|Direct Simulation
|-
|[[BB(5)]]
|<math>47\,176\,870</math>
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|Heiner Marxen & Jürgen Buntrock in 1989
|Direct Simulation
|-
|[[BB(6)]]
|<math>> 2\uparrow\uparrow\uparrow 5</math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|mxdys in 2025
|See mxdys's analysis on the TM page
|-
|[[BB(7)]]
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|[https://discord.com/channels/960643023006490684/1369339127652159509/1370678203395604562 Pavel Kropitz in 2025]
|Analyzed by Shawn Ligocki (see TM page)
|-
|[[BB(8)]]
|
|
|
|
|-
|BB(9)
|<math>> f_\omega(f_9(2))</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|Jacobzheng in 2024
|
|-
|BB(10)
|<math>> f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|Racheline in 2024
|
|-
|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}}
|Racheline in 2024
|
|-
|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}}
|Racheline in 2024
|
|-
|BB(13)
|<math>> f_{\omega + 1}(2047) > g_{64}</math>
|{{TM|1RB1RA_1LC1RD_1LA1LC_1LG0RE_1LC1RB_0RL1LG_0LM0RH_1RI1RH_1LK0RI_---0LK_1LF1LK_1LJ1RL_1RZ1RH|halt}}
|[https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 50_ft_lock in 2026]
|
|-
|BB(14)
|<math>> f_{\omega + 1}(65\,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Racheline in 2024]
|
|-
|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}}
|Jacobzheng in 2025
|
|-
|BB(16)
|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
|[[User:Jacobzheng/BB(16)]]
|Jacobzheng in 2025
|
|-
|BB(17)
|
|
|
|
|-
|BB(18)
|<math>> f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60)))</math>
|[[User:Jacobzheng/BB(18)]]
|Jacobzheng in 2025
|
|-
|BB(19)
|
|
|
|
|-
|BB(20)
|<math>> f_{\omega + 2}^2(21)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274414683331366924 Racheline in 2024]
|
|-
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|
|-
|BB(40)
|<math>> f_{\omega^\omega}(75\,500)</math>
|[[User:Jacobzheng/BB(40)]]
|Jacobzheng in 2024
|
|-
|BB(41)
|<math>> f_{\omega^\omega}^4(32)</math>
|[[User:Jacobzheng/BB(41)]]
|Jacobzheng in 2024
|
|-
|BB(51)
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Racheline in 2024]
|
|-
|BB(150)
|<math>f_{lim(BMS)}(10\uparrow\uparrow 15)</math>
|[https://morphett.info/turing/turing.html?c95a199c8e8a3dd56452f8b7e28fabbf too large to show]
|Patcail in 2025<ref>https://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345</ref>
|
|}

== 3-Symbol TMs ==

{| class="wikitable"
|+
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|[[BB(2,3)]]
|<math>38</math>
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|Allen Brady in 1988
|Direct Simulation
|-
|[[BB(3,3)]]
|<math>> 10^{17}</math>
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|Terry & Shawn Ligocki in 2007
|[https://bbchallenge.org/~pascal.michel/beh#tm33h Analysis by Pascal Michel]
|-
|[[BB(4,3)]]
|<math>> 10 \uparrow^{4} 4</math>
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|Pavel Kropitz in 2024
|
|}

== 4-Symbol TMs ==
{| class="wikitable"
|+
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|[[BB(2,4)]]
|<math>3\,932\,964</math>
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|Terry & Shawn Ligocki in 2005
|Pascal Michel, Heiner Marxen, Allen Brady
|-
|[[BB(3,4)]]
|<math>> 2 \uparrow^{15} 5</math>
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|Pavel Kropitz in 2024
|[https://www.sligocki.com/2024/05/22/bb-3-4-a14.html Analysis by Shawn Ligocki]
|}

== 5-Symbol TMs ==
{| class="wikitable"
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|[[BB(2,5)]]
|<math>> 10^{10^{10^{3\,314\,360}}}</math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|Daniel Yuan in 2024
|[https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 mxdys in Rocq]
|-
|[[BB(3,5)]]
|<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
|Racheline in 2024
|
|}

== 6-Symbol TMs ==
{| class="wikitable"
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|[[BB(2,6)]]
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|Pavel Kropitz in 2023
|[https://www.sligocki.com/2023/05/20/bb-2-6-p3.html Analysis by Shawn Ligocki]
|}

== Zoology ==
{| class="wikitable"
|+
!Classification
!Description
!Examples
!Scale
|-
|Trivial
|The simplest champions that can exist. They mostly appear in some BB-adjacent functions like [[Fractran|BBf]].
|
|<math>O(n)</math>
|-
|Chaotic
|Have a chaotic behavior with often repeating patterns that go back and forth.
|
* {{TM|1RB1LB_1LA---|halt}}
* {{TM|1RB---_1LB0RC_1LC1LA|halt}}
* {{TM|1RB1LB_1LA0LC_---1LD_1RD0RA|halt}}
|<math>O(n^2)</math>
|-
|Countdown
|Compute a number then "count down" (usually while bouncing) until reaching 0. They are common in some BB-adjacent functions like [[Fractran|BBf]].
|
* {{TM|1RB2LB---_2LA2RB1LB|halt}}
|<math>O(n^2)</math>
|-
|Collatz-like
|Compute a Collatz-like function. Repeatedly multiply and add a number depending of its modulo until reaching a number with a certain modulo.
|
* {{TM|1RB2LA1RA1RA_1LB1LA3RB---|halt}}
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_---0LA|halt}}
|<math>O(2^n)</math>
|}

== References ==

[[Category:Individual machines]]
[[Category:Zoology]]

Logical independence

2026-03-14T18:34:13Z

C7X: More common phrasing /* Large cardinals */

[[File:Independencechart.png|thumb|A chart showing which busy beaver numbers are independent of theories. Modification of a chart from a Vsauce video.]]
For any recursively enumerable and arithmetically sound axiomatic theory T, there exists an integer <math>N_T</math> such that T cannot prove the values of BB(n) for any <math>n \ge N_T</math>. For [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] (ZF), this value is known to be in the range:

<math display="block">6 \le N_{ZF} \le 432</math>

This lower bound comes from the fact that [[BB(5)]] has been proven in Rocq.<ref group="footnote">The fact that this theorem has been proven in Rocq technically does not imply that the theorem must be provable in ZF, because the consistency strength of Rocq is actually higher than that of ZF. However, the BB(5) proof does not use any techniques that could not be formalized within ZF.</ref> The upper bound comes from an explicit TM which enumerates all possible proofs in ZF and halts if it finds a proof 0 = 1. Assuming ZF is consistent and sound, then it cannot prove whether or not it is consistent, hence it cannot prove whether or not this specific TM halts.

Harvey Friedman spoke of embedding consistency statements within turing machines in a [https://fomarchive.ugent.be/2004-March/008003.html 2004 posting] on the "Foundations of Mathematics" mailing list. Scott Aaronson conjectured in his [[Busy Beaver Frontier]] survey that <math>N_{ZF} \le 20</math> and <math>N_{PA} \le 10</math>, where <math>PA</math> refers to the theory of [https://en.wikipedia.org/wiki/Peano%20axioms Peano Arithmetic]. Scott also mentioned (crediting Oscar Cunningham for the observation) that if there is an n-state machine that halts if and only if T is inconsistent, then <math>PA</math> + “BB(n) = b” can prove the consistency of T, where b is the true value of BB(n).<ref name=":0">Scott Aaronson. 2020. [https://www.scottaaronson.com/papers/bb.pdf The Busy Beaver Frontier]. SIGACT News 51, 3 (August 2020), 32–54. https://doi.org/10.1145/3427361.3427369</ref>

== Axiom of Choice ==
Due to [[wikipedia:Absoluteness_(logic)#Shoenfield's_absoluteness_theorem|Shoenfield's absoluteness theorem]], it is known that any TM proven non-halting in ZFC can also be proven non-halting in ZF (and the converse is trivially true), therefore

<math display="block">N_{ZFC} = N_{ZF}</math>

Therefore we refer to ZF and <math>N_{ZF}</math> throughout this article since adding the Axiom of Choice does not have any effect on Turing machine decidability.

== History ==
There is no one authoritative source on the history of TMs independent of ZF, this is our best understanding of the history of TMs found. Mostly these are taken from Scott Aaronson's blog announcements and Busy Beaver Frontier or self-reported by the individuals who discovered them. The Aaronson-Yedida machine used a compiler called ''Laconic'', which was then updated with NQL or "Not Quite Laconic", while the current champion machines use a compiler built by Andrew J. Wade.

Note that these results also have not undergone formal verification, with the 7910 machine in particular relying on a result from Harvey Friedman that has no published proof.
{| class="wikitable"
|+History of ZF independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|7910
|May 2016
|Adam Yedidia and Scott Aaronson
|Yedidia and Aaronson 2016<ref>A. Yedidia and S. Aaronson. A relatively small Turing machine whose behavior is independent of set theory. Complex Systems, (25):4, 2016. https://arxiv.org/abs/1605.04343</ref>
|
|-
|748
|May 2016
|Stefan O’Rear
|[https://github.com/sorear/metamath-turing-machines/blob/master/zf2.nql Github NQL file], Busy Beaver Frontier<ref name=":0" />
|
|-
|745
|July 2023
|Johannes Riebel
|Riebel 2023 Bachelor Thesis<ref>Riebel, Johannes (March 2023). ''[https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf The Undecidability of BB(748): Understanding Gödel's Incompleteness Theorems]'' (PDF) (Bachelor's thesis). University of Augsburg.</ref>
|
|-
|643
|July 2024
|Rohan Ridenour
|[https://github.com/CatsAreFluffy/metamath-turing-machines Github NQL], [https://scottaaronson.blog/?p=8131 Aaronson Announcement]
|
|-
|636
|31 August 2024
|Rohan Ridenour
|[https://github.com/CatsAreFluffy/metamath-turing-machines Github NQL] ([https://github.com/CatsAreFluffy/metamath-turing-machines/commit/6fc33bef6ba8885d26aed94c83e88bdabbedb0f1 commit])
|
|-
|588
|12 July 2025
|Andrew J. Wade
|[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/30d2e3194866615f68dd2f5101cb300fb039adca commit])
|
|-
|549
|16 July 2025
|Andrew J. Wade
|[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/5d676aec074a94f598959cb3b7733a8f7781762f commit])
|
|-
|432
|19 Aug 2025
|Andrew J. Wade
|[https://codeberg.org/ajwade/turing_machine_explorer/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e git commit]
|
|}

Wade's 432-state champion also omits the [[wikipedia:Axiom_of_regularity|axiom of regularity]].

For Peano Arithmetic, the ZF independent machines are an upper bound. In general, a machine that is independent of a theory will also be independent of any theory with strictly lower [[wikipedia:Equiconsistency#Consistency_strength|consistency strength]]. For the theories in this article, Con(ZFC+Subtle)->Con(ZFC)->Con(PA).

{| class="wikitable"
|+History of Peano Arithmetic (PA) independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|372
|11 February 2026
|@LegionMammal978
|[https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py Github]
|
|}
This machine takes Wade's ZF champion and replaces the set theory axioms with one axiom schema of "adjunction + separation", which has been shown to interpret PA.<ref>https://mathoverflow.net/questions/508137/interpreting-pa-with-only-adjunction-separation/508161</ref>

=== Large cardinals ===
These tables are for theories stronger than ZFC.

{| class="wikitable"
|+History of ZFC + "There exist arbitrarily large subtle cardinals" independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|493
|25 February 2026
|@LegionMammal978
|[https://github.com/Saoi2/turing_machine_explorer/blob/main/subtle.tm Github]
|
|}
This machine takes Wade's ZF champion and adds the [[wikipedia:Axiom_of_choice|axiom of Choice]] and a statement from Harvey Friedman (Proposition 4.3 of [https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf "Primitive Independence Results"])<ref>H. Friedman, "[https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf Primitive Independence Results]" (2002). Accessed February 2026.</ref> which Friedman showed to be equivalent to "There exist arbitrarily large subtle cardinals" over ZFC.

For reference, this theory is between the strength of "strongly unfoldable" and "<math>0^\sharp</math> exists" on the [[wikipedia:Large_cardinal|large cardinal hierarchy]] on Wikipedia.

Note: The initial machine produced in 2016 by Yedidia and Aaronson is designed to halt iff a certain statement created by Harvey Friedman is true.<ref name=":0" /> According to Friedman (without a published proof), this statement is independent of the theory "Stationary Ramsey Property" or ''SRP'', which is equiconsistent with the theory <math>ZFC + \{\text{``There is a }k\text{-subtle cardinal``}\mid k\in\N\}</math> which is also between the strength of "strongly unfoldable" and "<math>0^\sharp</math> exists" on the large cardinal hierarchy.<ref>tlonuqbar, "[https://mathoverflow.net/questions/508364/what-is-the-consistency-strength-of-srp-stationary-ramsey-property What is the consistency strength of SRP (Stationary Ramsey Property)?]" (2026). MathOverflow post, accessed February 2026.</ref> However, the theory used by the BB(493) machine is also independent of this theory.

== Footnotes ==
<references group="footnote"/>

== References ==
<references />

[[Category:Zoology]]

BB(7)

2026-03-14T08:50:44Z

C7X: Done /* History */

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

The current BB(7) [[champion]] {{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}} was discovered by Pavel Kropitz in May 2025, proving the lower bound: <math display="block">S(7) > \Sigma(7) > 2 \uparrow^{11} 2 \uparrow^{11} 3</math>.

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

In May 2025, mxdys shared [https://github.com/ccz181078/TM C++ code] that breaks up the BB(7) enumeration into 1 million subtasks which each ran for ~2 minutes and leave ~100 [[Holdouts lists|holdouts]] each. Various folks on Discord investigated different sections of this domain to search for champions.

Within three days of the code's release, the Ligockis found three champions after applying their deciders to enumerator output. Shawn Ligocki found the first two, {{TM|1RB0RF_1LC0RE_1RD1LB_1LA1LD_0RA0LE_1RG0LB_1RZ1RB|halt}} and {{TM|1RB1RA_1RC0LC_0LD1LG_1LF0LE_1RZ1LF_0LA1LD_1RA1LC|halt}}, with sigma scores of approximately 10 ↑↑ 22 and 10 ↑↑ 35. That evening, Terry Ligocki found {{TM|1RB0LG_1RC0RF_1LD1RZ_1LF0LE_1RA1LD_1LG1RE_0LB0LB|halt}}, with sigma score ~10 ↑↑ 46. A few days later, Pavel found a TM that outpaces all of them with a sigma score of ~<math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>. Pavel's champion is enumerated in subtask 243308 of Phase 1 (below).

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

[[Probviously]] non-halting Cryptids:

{{TM|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB}}, [[Bigfoot]] (a [[BB(3,3)]] Cryptid) compiled into a 2-symbol TM by Iijil in 2024.

== 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 top 20 scoring known machines are:
{| class="wikitable sortable"
!TM
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|data-sort-value="10 ↑↑ 9999"|<math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|Pavel Kropitz
|-
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE|halt}}
|10 ↑↑ 519.20
|Andrew Ducharme
|-
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE|halt}}
|10 ↑↑ 519.20
|Andrew Ducharme
|-
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD0LG_1LB0LE|halt}}
|10 ↑↑ 519.20
|@gerbil5709, Terry Ligocki
|-
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG|halt}}
|10 ↑↑ 403.84
|Andrew Ducharme
|-
|{{TM|1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ|halt}}
|10 ↑↑ 286.17
|Terry Ligocki
|-
|{{TM|1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA0LG_1RZ0RD|halt}}
|10 ↑↑ 246.32
|@Iijil
|-
|{{TM|1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA1LG_1RZ1LA|halt}}
|10 ↑↑ 246.32
|@star, Terry Ligocki
|-
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG|halt}}
|10 ↑↑ 243.88
|@Iijil, Andrew Ducharme
|-
|{{TM|1RB0RB_1LC1RG_1RD1RC_1RE0RA_1LF0LB_1RF0LE_0RD1RZ|halt}}
|10 ↑↑ 228.78
|Terry Ligocki
|-
|{{TM|1RB0LD_0LC1RZ_1RA0RD_1RE1LD_1LF0RC_0LG1LE_1RG0LD|halt}}
|10 ↑↑ 192.67
|Terry Ligocki
|-
|{{TM|1RB1LA_1LC0RE_0LD1LB_1RD0LA_1RF0RA_0RG0LA_1RB1RZ|halt}}
|10 ↑↑ 192.67
|Terry Ligocki
|-
|{{TM|1RB1LA_1LC0RE_0LD1LB_1RD0LA_1RF0RA_1RG0LA_0LE1RZ|halt}}
|10 ↑↑ 192.67
|Terry Ligocki
|-
|{{TM|1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RA0LG_1RB1LG|halt}}
|10 ↑↑ 192.67
|Terry Ligocki
|-
|{{TM|1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RG0LG_1RB1LG|halt}}
|10 ↑↑ 192.67
|Terry Ligocki
|-
|{{TM|1RB0LD_0LC1RZ_1RA0RD_1RE1LD_1LF0RC_0LG1LE_1RC0LD|halt}}
|10 ↑↑ 192.67
|Andrew Ducharme
|-
|{{TM|1RB1LA_1LC0RE_0LD1LB_1RE1LG_1RF0LG_0RA0LA_0RF1RZ|halt}}
|10 ↑↑ 192.67
|Andrew Ducharme
|-
|{{TM|1RB1LA_1LC0RF_0LD0RD_1RF1LE_1LB1RZ_1RG0RA_0RA0LA|halt}}
|10 ↑↑ 192.67
|@C7X
|-
|{{TM|1RB1LA_1LC0RE_0LD1LB_1RE0LA_1RF0RA_0RG0LA_1RB1RZ|halt}}
|10 ↑↑ 192.67
|@Iijil, Terry Ligocki
|-
|{{TM|1RB1LA_1LC0RE_0LD1LB_1RE0LA_1RF0RA_1RG0LA_0LE1RZ|halt}}
|10 ↑↑ 192.67
|@Iijil, Terry Ligocki
|}

The top 20 known halters with unique scores are:

{| class="wikitable sortable"
!TM
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|data-sort-value="10 ↑↑ 9999"|<math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|Pavel Kropitz
|-
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE|halt}}
|10 ↑↑ 519.20
|Andrew Ducharme
|-
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG|halt}}
|10 ↑↑ 403.84
|Andrew Ducharme
|-
|{{TM|1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ|halt}}
|10 ↑↑ 286.17
|Terry Ligocki
|-
|{{TM|1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA0LG_1RZ0RD|halt}}
|10 ↑↑ 246.32
|@Iijil
|-
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG|halt}}
|10 ↑↑ 243.88
|@Iijil, Andrew Ducharme
|-
|{{TM|1RB0RB_1LC1RG_1RD1RC_1RE0RA_1LF0LB_1RF0LE_0RD1RZ|halt}}
|10 ↑↑ 228.78
|Terry Ligocki
|-
|{{TM|1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RA0LG_1RB1LG|halt}}
|10 ↑↑ 192.67
|Terry Ligocki
|-
|{{TM|1RB0LC_1LC1LD_1LA1LB_0LG1RE_1LD0RF_0RA1RE_1RZ1LA|halt}}
|10 ↑↑ 188.28
|Terry Ligocki
|-
|{{TM|1RB0LC_1LC0LD_1LA1LB_0LG1RE_1LD0RF_0RA1RE_1RZ1LC|halt}}
|10 ↑↑ 140.28
|@stokastic
|-
|{{TM|1RB0RF_1RC1RZ_0LD1RF_0RA1LE_0LC1LF_1LE0RG_0LE1RA|halt}}
|10 ↑↑ 136.64
|Katelyn Doucette, Andrew Ducharme
|-
|{{TM|1RB0LG_0RC1RZ_1LD0LA_1RE1LE_1LC1RF_0RE0RA_0RF1LG|halt}}
|10 ↑↑ 133.85
|@poppuncher
|-
|{{TM|1RB1RZ_1RC0RF_1LD1RB_1RG0LE_1LD0RA_1RE0LD_0RC1LF|halt}}
|10 ↑↑ 129.24
|@Iijil
|-
|{{TM|1RB0RG_1LC0RE_1LF1LD_0LE1LC_1RA1RB_1LD0LF_1RZ0RF|halt}}
|10 ↑↑ 127.52
|Andrew Ducharme
|-
|{{TM|1RB0LC_1RC0RG_1RD0LF_1RE0RF_1LA1RG_1LE1LF_1RZ1RD|halt}}
|10 ↑↑ 126.20
|@stokastic
|-
|{{TM|1RB0LD_1RC1RA_0RD1RG_1LE1LF_0LF1LE_0RA0LD_0RA1RZ|halt}}
|10 ↑↑ 124.86
|Terry Ligocki
|-
|{{TM|1RB1LF_1RC1RA_1LD0LD_1LA1LE_0LA0LD_1LG0RF_0LE1RZ|halt}}
|10 ↑↑ 116.98
|Terry Ligocki
|-
|{{TM|1RB0RD_1RC0LA_0LA0LE_1RE1RZ_1RF0RA_1LG0LE_1LC0LG|halt}}
|10 ↑↑ 116.05
|
|-
|{{TM|1RB0RD_1RC0RA_0RD1LD_0LE1LF_1LA0LG_0LC1LB_1LC1RZ|halt}}
|10 ↑↑ 115.52
|@prurq
|-
|{{TM|1RB0RG_1LC0LE_1LD0LB_0LE1RE_0RA1RF_0RD1RC_1RD1RZ|halt}}
|10 ↑↑ 114.83
|Andrew Ducharme
|}

== Phase 1 ==
Phase 1's first two stages were carried out from May 2025 to July 2025. The table below summaries some of that activity and was used to coordinate the effort of doing the raw computation. Fourteen people (see below) contributed directly and the overall bbchallenge group participated in [https://discord.com/channels/960643023006490684/1369339127652159509 discord discussions], etc.

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

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

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

(done to reduce column size:
<math>*^1</math>= enumeration,
<math>*^2</math>= linear rule)

{| class="wikitable sortable"
! rowspan="2" |Task range
! rowspan="2" |Done by
! colspan="2" |Completed
! rowspan="2" |# holdouts
! rowspan="2" |Maximum Score TM
! rowspan="2" |~Sigma
! rowspan="2" |Source
|-
!<math>*^1</math>
!<math>*^2</math>
|-
|00-01xxxx
|@Iijil
|Yes
|Yes
|1,545,673
|{{TM|1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA0LG_1RZ0RD|halt}}
|10 ↑↑ 246.32
|[https://drive.google.com/drive/folders/1wniwrAuvsHfkvro8Tg65WAMNZEuIekzD Google Drive folder]
|-
|02-04xxxx
|
@Iijil<br/>
Terry Ligocki
|Yes
|Yes
|2,279,734
|{{TM|1RB0LF_1RC1RA_1RD0RG_1LE1RZ_1LA0LF_1RA1LE_0RE1RG|halt}}
|10 ↑↑ 93.81
|
[https://drive.google.com/drive/folders/1wniwrAuvsHfkvro8Tg65WAMNZEuIekzD @Iijil]<br/>
[https://drive.google.com/drive/folders/1kJ6tlmX8_7AQ8qpR1mSQ-Fz4-fPfwYBn?usp=drive_link Terry Ligocki]
|-
|05-09xxxx
|
@Iijil<br/>
Andrew Ducharme
|Yes
|Yes
|3,889,955
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG|halt}}
|10 ↑↑ 243.88
|
[https://drive.google.com/drive/folders/1wniwrAuvsHfkvro8Tg65WAMNZEuIekzD @Iijil]<br/>
[https://drive.google.com/drive/folders/16uDjgOahkhAMWv3v-YWmxJG7xxsBvj4h?usp=sharing Andrew]
|-
|10-12xxxx
|Andrew Ducharme
|Yes
|Yes
|2,708,888
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE|halt}}
|10 ↑↑ 519.20
|[https://drive.google.com/drive/folders/16uDjgOahkhAMWv3v-YWmxJG7xxsBvj4h?usp=sharing Google Drive folder]
|-
|13xxxx
|Shawn Ligocki
|Yes
|Yes
|1,192,442
|{{TM|1RB0RE_1LC0LA_1LD0LC_0LE0LA_1RF0RG_1RD0LE_1RA1RZ|halt}}
|10 ↑↑ 114.60
|[https://drive.google.com/drive/folders/1lyYN2wznnrfM0dg-dKprHODeYaTxdtzP?usp=drive_link Google Drive folder]
|-
|14-16xxxx
|Andrew Ducharme
|Yes
|Yes
|2,701,637
|{{TM|1RB0LC_1LC1LD_1LA1LB_0LG1RE_0RF0LD_0RA1RE_1RZ1LA|halt}}
|10 ↑↑ 188.28
|[https://drive.google.com/drive/folders/16uDjgOahkhAMWv3v-YWmxJG7xxsBvj4h?usp=sharing Google Drive folder]
|-
|17-18xxxx
|
@gerbil5709<br/>
Terry Ligocki
|Yes
|Yes
|1,898,156
|{{TM|1RB1LA_1LC0RE_0LD1LB_1RE0LA_1RF0RA_0RG0LA_1RB1RZ|halt}}
|10 ↑↑ 192.67
|
[https://drive.google.com/drive/folders/1kAvBebeF09CEVocCk5bGKlDJfRN8co_i?usp=sharing @gerbil5709]<br/>
[https://drive.google.com/drive/folders/1kJ6tlmX8_7AQ8qpR1mSQ-Fz4-fPfwYBn?usp=drive_link Terry Ligocki]
|-
|19xxxx
|
Katelyn Doucette<br/>
Andrew Ducharme
|Yes
|Yes
|1,099,752
|{{TM|1RB0RF_1RC1RZ_0LD1RF_0RA1LE_0LC1LF_1LE0RG_0LE1RA|halt}}
|10 ↑↑ 136.64
|[https://drive.google.com/drive/folders/1-eGxVc3kmGIEJFShG4olPX3sGci2SPaA?usp=sharing Google Drive folder]
|-
|20-23xxxx
| @C7X
|Yes
|Yes
|4,528,827
|{{TM|1RB1LA_1LC0RF_0LD0RD_1RF1LE_1LB1RZ_1RG0RA_0RA0LA|halt}}
|10 ↑↑ 192.67
| [https://drive.google.com/drive/folders/11iGTKsvu2Y7aFrwOcWS1LYvcN6i_7-JM?usp=sharing Google Drive folder]
|-
|24xxxx
|Andrew Ducharme
|Yes
|Yes
|712,356
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}*
|data-sort-value="10 ↑↑ 9999"|<math>2 \uparrow^{11} 2 \uparrow^{11} 3^*</math>
|[https://drive.google.com/drive/folders/16uDjgOahkhAMWv3v-YWmxJG7xxsBvj4h?usp=sharing Google Drive folder]
|-
|25-34xxxx
|@stokastic
|Yes
|Yes
|10,339,816
|{{TM|1RB0LC_1LC0LD_1LA1LB_0LG1RE_1LD0RF_0RA1RE_1RZ1LC|halt}}
|10 ↑↑ 140.28
|[https://drive.google.com/drive/folders/16_qIdWWD-wolj6zURB5ZSbY-otI4zoUF?usp=sharing Google Drive folder]
|-
|35-39xxxx
|Terry Ligocki
|Yes
|Yes
|4,894,047
|{{TM|1RB1RZ_1LC0RF_0LD1LB_1RD0LE_1RB1LE_1RG0RE_0RA0LE|halt}}
|10 ↑↑ 192.67
|[https://drive.google.com/drive/folders/1kJ6tlmX8_7AQ8qpR1mSQ-Fz4-fPfwYBn?usp=drive_link Google Drive folder]
|-
|40-47xxxx
|Andrew Ducharme
|Yes
|Yes
|6,181,327
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE|halt}}
|10 ↑↑ 519.20
|[https://drive.google.com/drive/folders/16uDjgOahkhAMWv3v-YWmxJG7xxsBvj4h?usp=sharing Google Drive folder]
|-
|48xxxx
|
@star<br/>
Terry Ligocki
|Yes
|Yes
|727,875
|{{TM|1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA1LG_1RZ1LA|halt}}
|10 ↑↑ 246.32
|
[https://drive.google.com/file/d/1HbIX46_6V-etFWTv4FvWZmb7AHIiWB1v/view?usp=sharing @star]<br/>
[https://drive.google.com/drive/folders/1lyYN2wznnrfM0dg-dKprHODeYaTxdtzP?usp=drive_link Terry Ligocki]
|-
|49xxxx
|
Tobiáš Brichta<br/>
Terry Ligocki
|Yes
|Yes
|804,722
|{{TM|1RB0LG_1RC0RG_0LD1RE_1RD0RE_1LF1RB_0LA1RZ_1LC1LG|halt}}
|10 ↑↑ 126.20
|
[https://drive.google.com/drive/folders/1-csgJ5uSIX3SKlqTkSnhkUuEYLKgCw81 Tobiáš Brichta]<br/>
[https://drive.google.com/drive/folders/1kJ6tlmX8_7AQ8qpR1mSQ-Fz4-fPfwYBn?usp=drive_link Terry Ligocki]
|-
|50xxxx
|
@prurq<br/>
Andrew Ducharme
|Yes
|Yes
|797,224
|{{TM|1RB0RD_1RC0RA_0RD1LD_0LE1LF_1LA0LG_0LC1LB_1LC1RZ|halt}}
|10 ↑↑ 115.52
|[https://drive.google.com/drive/folders/145H4sT4F9KJYGSrlIETZdBOIMR7krLQm Google Drive folder]
|-
|51-53xxxx
|
@gerbil5709<br/>
Terry Ligocki
|Yes
|Yes
|3,016,175
|{{TM|1RB0LC_1LC0LD_1LA1LB_0LG1RE_0RF0RF_0RA1RE_1RZ1LC|halt}}
|10 ↑↑ 140.28
|
[https://drive.google.com/drive/folders/1kAvBebeF09CEVocCk5bGKlDJfRN8co_i?usp=sharing @gerbil5709]<br/>
[https://drive.google.com/drive/folders/1kJ6tlmX8_7AQ8qpR1mSQ-Fz4-fPfwYBn?usp=drive_link Terry Ligocki]
|-
|54-59xxxx
|Terry Ligocki
|Yes
|Yes
|5,689,850
|{{TM|1RB0LC_1LC1LD_1LA1LB_0LG1RE_0RF0RF_0RA1RE_1RZ1LA|halt}}
|10 ↑↑ 188.28
|[https://drive.google.com/drive/folders/1lyYN2wznnrfM0dg-dKprHODeYaTxdtzP?usp=drive_link Google Drive folder]
|-
|60-64xxxx
|
@gerbil5709<br/>
Terry Ligocki
|Yes
|Yes
|3,817,876
||{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD0LG_1LB0LE|halt}}
|10 ↑↑ 519.20
|
[https://drive.google.com/drive/folders/1kAvBebeF09CEVocCk5bGKlDJfRN8co_i?usp=sharing @gerbil5709]<br/>
[https://drive.google.com/drive/folders/1kJ6tlmX8_7AQ8qpR1mSQ-Fz4-fPfwYBn?usp=drive_link Terry Ligocki]
|-
|65-68xxxx
|Terry Ligocki
|Yes
|Yes
|3,076,778
|{{TM|1RB0LD_0LC1RZ_1RA0RD_1RE1LD_1LF0RC_0LG1LE_1RG0LD|halt}}
|10 ↑↑ 192.67
|[https://drive.google.com/drive/folders/1lyYN2wznnrfM0dg-dKprHODeYaTxdtzP?usp=drive_link Google Drive folder]
|-
|69xxxx
|@poppuncher
|Yes
|Yes
|1,053,119
|{{TM|1RB0LG_0RC1RZ_1LD0LA_1RE1LE_1LC1RF_0RE0RA_0RF1LG|halt}}
|10 ↑↑ 133.85
|[https://drive.google.com/drive/folders/1KlCZqXxqVPuBPkDcCBocuMPA8paq9b8P?usp=drive_link Google Drive folder]
|-
|70-71xxxx
|@hipparcos
|Yes
|Yes
|1,899,094
|{{TM|1RB1RZ_1LC1RD_0LD0LC_1LE1RA_1LF0LE_1RF0RG_1RG0RD|halt}}
|10 ↑↑ 77.50
|[https://github.com/jhuang97/bb7x2/releases Github release]
|-
|72-79xxxx
|Terry Ligocki
|Yes
|Yes
|7,627,514
|{{TM|1RB0RB_1LC1RG_1RD1RC_1RE0RA_1LF0LB_1RF0LE_0RD1RZ|halt}}
|10 ↑↑ 228.78
|[https://drive.google.com/drive/folders/1kJ6tlmX8_7AQ8qpR1mSQ-Fz4-fPfwYBn?usp=drive_link Google Drive folder]
|-
|80-81xxxx
|[[User:XnoobSpeakable|XnoobSpeakable]]
|Yes
|Yes
|1,537,533
|{{TM|1RB0LA_0RC1RZ_0RD0RG_1LE1RA_1LF1LD_1RG0RG_1RD1RC|halt}}
|10 ↑↑ 74.85
|[https://drive.google.com/drive/folders/1TpuEC7KottEmvsFnCREugnlVMPaY5ZHi?usp=sharing Google Drive folder]
|-
|82-99xxxx
|Terry Ligocki
|Yes
|Yes
|15,673,786
|{{TM|1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ|halt}}
|10 ↑↑ 286.17
|[https://drive.google.com/drive/folders/1kJ6tlmX8_7AQ8qpR1mSQ-Fz4-fPfwYBn?usp=drive_link Google Drive folder]
|}
<nowiki>*</nowiki>The current BB(7) champion TM {{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}* was discovered by Pavel Kropitz in the enumeration of subtask 243308. The remaining subtasks in the 24xxxx range were enumerated and filtered by Andrew Ducharme.

== Exploration after Phase 1 ==
People are now looking at reducing the number of holdouts (~85M TMs) after Phase 1. They are trying different deciders not used in Phase 1 and different parameters with the deciders used in Phase 1. This table is somewhere they can put the results of these explorations so that there is a record of what is being done. It is hoped that this will inspire everyone to contribute when they can, a reference point for [https://discord.com/channels/960643023006490684/1369339127652159509 discord discussions], and something that can be questioned and/or verified.

(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"
! style="width: 10%" rowspan="2" |Done by
! colspan="2" |Holdout TMs
! rowspan="2" |<math>*^1</math>
! rowspan="2" |<math>*^2</math>
! colspan="2" |TMs/sec/core
! style="width: 50%" rowspan="2" |Description
! rowspan="2" |Source
|-

!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:center" |Shawn Ligocki
|858,538
|733,830
|14.5%
|5.0
|6.9282
|47.6966
|style="text-align:left" |[[Translated Cycler]] and [[CPS]] on <code>7x2_p01_s02_holdouts_rand_13.txt</code>
|style="text-align:left" |[https://discord.com/channels/960643023006490684/1369339127652159509/1403423336293208164 Discord link]
|-
|style="text-align:center" |Andrew Ducharme
|872,041
|784,099
|10.1%
|456.4
|0.0535
|0.5034
|style="text-align:left" |Enumerate.py w/ 250k max-loops and block-mult=3 on <code>7x2_p01_s02_holdouts_rand_72.txt</code>
|style="text-align:left" |[https://discord.com/channels/960643023006490684/1369339127652159509/1403236957298884679 Discord link]
|-
|style="text-align:center" |Terry Ligocki
|1,000
|982
|1.8%
|3.3
|0.0015
|0.0830
|style="text-align:left" |[[MITMWFAR]] [https://github.com/Iijil1/MITMWFAR Code] with options <code>-n=10 -m=1 -pm=1</code> on the first 1,000 TMs in <code>7x2_p01_s03_holdouts_rand_47.txt</code>
|style="text-align:left" |
|-
|style="text-align:center" |Andrew Ducharme
|800,507
|693,348
|13.3%
|2818.7
|0.0106
|0.0789
|style="text-align:left" |Enumerate.py w/ 1M max-loops and block-mult=4 on holdouts from above Enumerate run on<code>7x2_p01_s02_holdouts_rand_72.txt</code>
|style="text-align:left" |
|-
|style="text-align:center" |Andrew Ducharme
|70,000
|65,615
|6.3%
|6.6
|0.1842
|2.9412
|style="text-align:left" |Enumerate.py with --no-sim and --lin-steps=100000 on <code>7x2_p01_s02_holdouts_rand_65.txt</code>
|style="text-align:left" |
|-
|style="text-align:center" |Terry Ligocki
|10,000
|N/A
|2% - 32%
|N/A
|N/A
|N/A
|style="text-align:left" |See the "Source" discussion on the discord channel which includes graphs with quantitative data. A parameter study of a random sample of 10,000 TMs from the Phase 1, Stage 3 holdouts
|style="text-align:left" |[https://discord.com/channels/960643023006490684/1369339127652159509/1407090465097908245 Discord link]
|}

== Phase 2 ==
Phase 2 began with Andrew Ducharme's initial decider runs (see Stage 1 in the table). The goal of Phase 2 is to run deciders on the current holdout list (the last in the table) and produce an accessible new holdout list which can then be added to the table. Input Holdout TMs may be unequal to the next row's Output Holdout TMs because more TMs are TNF-enumerated in the process of executing the next row's filtering. Accessing the holdout list can be done through the link "Holdouts" in the "Source" column. The link "Details" leads to additional data/information pertaining to each step. If there isn't a link there it means the data is still being put somewhere everyone can access it.

Stage 1 reduced the number of holdouts by 33.3% from ~86.1M to ~57.5M TMs using the Ligocki's C++/Python codes. In Stage 2, Andrew continued using mxdys' C++ code and reduced the number of holdouts by 51.0% from ~57.5M to ~28.2M TMs. Terry Ligocki then pushed this further using additional deciders/parameters to generate Stage 3 which reduced the number of holdouts by 17.3% from ~28.2M to ~23.3M TMs. Andrew then ran the Ligocki's C++/Python code with larger parameters which reduced the number of holdouts by 2.6% from ~23.3M to ~22.7M TMs. This formed Stage 4. Terry then switched back to the mxdys' C++ code with different deciders/parameters for Stage 5 where the number of holdouts was reduced by 10.2% from ~22.7M to ~20.4M TMs.

This brought the overall reduction in Phase 2 to 77.0% from ~86.1M to ~19.8M TMs. The details are contained in this table:

(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" |Source
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|colspan="20" style="text-align:center"|'''Stage 1'''
|-
|style="text-align:center" |Andrew Ducharme
|86,129,304
|82,226,951
|4.5%
|119.6
|9.06
|200.04
|style="text-align:left" |CPS_Filter with --max-block-size=4
|[https://discord.com/channels/960643023006490684/1369339127652159509/1407167730121052231 discord]
|rowspan="11" style="text-align:left" |Stage 1<br>
[https://drive.google.com/file/d/1LXUKxqRhwW_Q1QGxWuRheV3sZu18hgOK/view?usp=drive_link Holdouts]<br>
[https://drive.google.com/drive/u/0/folders/17U0BRpJHTMLtB0poBlOSZhGGp4FkCHIO Details]
|-
|style="text-align:center" |Andrew Ducharme
|82,226,951
|73,751,624
|10.3%
|120.0
|19.62
|190.34
|style="text-align:left" |lr_enum_continue 1M steps
|[https://discord.com/channels/960643023006490684/1369339127652159509/1407809787734786159 discord]
|-
|style="text-align:center" |Andrew Ducharme
|73,751,624
|72,470,054
|1.7%
|1040.0
|0.34
|19.70
|style="text-align:left" |lr_enum_continue 3M steps
|[https://discord.com/channels/960643023006490684/1369339127652159509/1408119196931330190 discord]
|-
|style="text-align:center" |Andrew Ducharme
|72,470,054
|69,347,610
|4.3%
|1113.4
|0.78
|18.08
|style="text-align:left" |CPS_Filter with --min-block-size=5, --max-block-size=6
|[https://discord.com/channels/960643023006490684/1369339127652159509/1408517862485917826 discord]
|-
|style="text-align:center" |Andrew Ducharme
|69,347,610
|68,695,205
|0.9%
|1491.8
|0.12
|12.92
|style="text-align:left" |CPS_Filter with --block-size=7
|[https://discord.com/channels/960643023006490684/1369339127652159509/1409227033715933244 discord]
|-
|style="text-align:center" |Andrew Ducharme
|68,695,205
|61,875,401
|10.6%
|1726.7
|1.17
|11.11
|style="text-align:left" |Enumerate.py with --block-multiple=12, max-loops=100_000, and --time=0.1
|[https://discord.com/channels/960643023006490684/1369339127652159509/1409590262644215828 discord]
|-
|style="text-align:center" |Andrew Ducharme
|61,875,401
|60,986,231
|1.5%
|1660.7
|0.15
|10.35
|style="text-align:left" |Enumerate.py with --block-multiple=8, max-loops=100_000, and --time=0.1
|[https://discord.com/channels/960643023006490684/1369339127652159509/1409939131701792788 discord]
|-
|style="text-align:center" |Andrew Ducharme
|60,986,231
|60,765,943
|0.4%
|1619.4
|0.04
|10.46
|style="text-align:left" |Enumerate.py with --block-multiple=16, max-loops=100_000, and --time=0.1
|[https://discord.com/channels/960643023006490684/1369339127652159509/1413962201676517396 discord]
|-
|style="text-align:center" |Andrew Ducharme
|60,765,943
|59,727,905
|1.7%
|2329.4
|0.12
|7.25
|style="text-align:left" |CPS_Filter with --block-size=8
|[https://discord.com/channels/960643023006490684/1369339127652159509/1411505653599572008 discord]
|-
|style="text-align:center" |Andrew Ducharme
|59,727,905
|57,452,672
|3.9%
|2472.2
|0.26
|6.71
|style="text-align:left" |Enumerate.py with --block-multiple=5, max-loops=200_000, and time=0.2
|[https://discord.com/channels/960643023006490684/1369339127652159509/1412882390774321172 discord]
|-
|style="text-align:center" |'''Stage 1 Cumulative'''
|'''86,129,304'''
|'''57,452,672'''
|'''33.3%'''
| '''13693.2'''
| ---
| ---
|style="text-align:left" | ---
|style="text-align:center" | ---
|-
|colspan="20" style="text-align:center"|'''Stage 2'''
|-
|style="text-align:center" |Andrew Ducharme
|57,452,672
|52,605,872
|8.4%
|150.0
|8.98
|106.39
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3
|[https://discord.com/channels/960643023006490684/1369339127652159509/1413022512568074341 discord]
|rowspan=21" style="text-align:left" |Stage 2<br>
[https://drive.google.com/file/d/18G2ofUaMZIKFwNCNVHxTRjasV6p39Wr2/view?usp=drive_link Holdouts]<br>
[https://drive.google.com/drive/folders/1tdJVC0OvUF8-Ql__xPKoSHKm5Lf5VoUd?usp=drive_link Details]
|-
|style="text-align:center" |Andrew Ducharme
|52,605,872
|50,268,427
|4.4%
|100.0
|6.49
|146.13
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 10000 NG_n 3
|[https://discord.com/channels/960643023006490684/1369339127652159509/1413589351933153352 discord]
|-
|style="text-align:center" |Andrew Ducharme
|50,268,427
|45,980,438
|8.5%
|250.0
|
|
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 8
|[https://discord.com/channels/960643023006490684/1369339127652159509/1413652363599810660 discord]
|-
|style="text-align:center" |Andrew Ducharme
|45,980,438
|43,870,806
|4.5%
|220.0
|
|
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 8 mod 3 n 6
|[https://discord.com/channels/960643023006490684/1369339127652159509/1413942432533446706 discord]
|-
|style="text-align:center" |Andrew Ducharme
|43,870,806
|42,700,370
|2.6%
|80.0
|
|
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 30000 NG_n 2
|[https://discord.com/channels/960643023006490684/1369339127652159509/1413942432533446706 discord]
|-
|style="text-align:center" |Andrew Ducharme
|42,700,370
|41,926,200
|1.8%
|20.0
|
|
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 3 n 1
|[https://discord.com/channels/960643023006490684/1369339127652159509/1413942432533446706 discord]
|-
|style="text-align:center" |Andrew Ducharme
|41,926,200
|41,590,605
|0.8%
|50.0
|
|
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 6
|[https://discord.com/channels/960643023006490684/1369339127652159509/1413942432533446706 discord]
|-
|style="text-align:center" |Andrew Ducharme
|41,590,605
|40,481,477
|2.6%
|75.0
|
|
|style="text-align:left" |chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2
|[https://discord.com/channels/960643023006490684/1369339127652159509/1413942432533446706 discord]
|-
|style="text-align:center" |Andrew Ducharme
|40,481,477
|38,641,627
|4.5%
|200.0
|
|
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n 12
|[https://discord.com/channels/960643023006490684/1369339127652159509/1413942432533446706 discord]
|-
|style="text-align:center" |Andrew Ducharme
|38,641,627
|37,514,197
|2.9%
|80.0
|
|
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 10000 NG_n 5
|[https://discord.com/channels/960643023006490684/1369339127652159509/1413966380377833552 discord]
|-
|style="text-align:center" |Andrew Ducharme
|37,514,197
|36,273,782
|3.3%
|90.0
|
|
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 1 n 4
|[https://discord.com/channels/960643023006490684/1369339127652159509/1414167850180018317 discord]
|-
|style="text-align:center" |Andrew Ducharme
|36,273,782
|35,984,179
|0.8%
|20.0
|
|
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 2 tH 0 n 2
|[https://discord.com/channels/960643023006490684/1369339127652159509/1414167850180018317 discord]
|-
|style="text-align:center" |Andrew Ducharme
|35,984,179
|31,811,445
|11.6%
|800.0
|
|
|style="text-align:left" |chr_LRUH 24 chr_H 24 MitM_CTL NG maxT 100000 NG_n 8
|[https://discord.com/channels/960643023006490684/1369339127652159509/1414167850180018317 discord]
|-
|style="text-align:center" |Andrew Ducharme
|31,811,445
|30,638,201
|3.6%
|1150.0
|
|
|style="text-align:left" |chr_LRUH 28 chr_H 28 MitM_CTL NG maxT 100000 NG_n 10
|[https://discord.com/channels/960643023006490684/1369339127652159509/1414385826099495004 discord]
|-
|style="text-align:center" |Andrew Ducharme
|30,638,201
|29,781,771
|2.8%
|630.0
|
|
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 12 mod 2 n 12
|[https://discord.com/channels/960643023006490684/1369339127652159509/1414654007149989978 discord]
|-
|style="text-align:center" |Andrew Ducharme
|29,781,771
|29,670,310
|0.37%
|30.0
|
|
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n [1-7]
| rowspan="4" |[https://discord.com/channels/960643023006490684/1369339127652159509/1416506454965227773 discord]
|-
|style="text-align:center" |Andrew Ducharme
|29,670,310
|29,629,503
|0.14%
|75.0
|
|
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n [8-10]
|-
|style="text-align:center" |Andrew Ducharme
|29,629,503
|29,380,949
|0.84%
|90.0
|
|
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 6
|-
|style="text-align:center" |Andrew Ducharme
|29,380,949
|28,543,434
|2.85%
|600.0
|
|
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n [2-10,2]
|-
|style="text-align:center" |Andrew Ducharme
|28,543,434
|28,189,617
|1.24%
|550.0
|
|
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n [1-11,2]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1417199742361927690 discord]
|-
| style="text-align:center" |'''Stage 2 Cumulative'''
|'''57,452,672'''
|'''28,189,617'''
|'''51.00%'''
| '''5260.0'''
| ---
| ---
| style="text-align:left" | ---
| style="text-align:center" | ---
|-
|colspan="20" style="text-align:center"|'''Stage 3'''
|-
|style="text-align:center" |Terry Ligocki
|28,189,617
|28,109,540
|0.28%
|5.3
|4.2
|1,479.75
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 3 n 1 run
|
|rowspan="25" style="text-align:left" |Stage 3<br>
[https://drive.google.com/file/d/1VEC5hum9Z9nkDhOhAw1bWsS8yLalrGWC/view?usp=drive_link Holdouts]<br>
[https://drive.google.com/drive/folders/1DPO_aJ25bqHYB6zzjPYkvnhEtVGsY9iZ?usp=drive_link Details]
|-
|style="text-align:center" |Terry Ligocki
|28,109,540
|27,804,922
|1.08%
|8.3
|10.18
|938.98
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 2 mod 6 n 1 run
|
|-
|style="text-align:center" |Terry Ligocki
|27,804,922
|27,747,435
|0.21%
|14.9
|1.07
|518.17
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 3 n 1 run
|
|-
|style="text-align:center" |Terry Ligocki
|27,747,435
|27,616,006
|0.47%
|65.7
|0.56
|117.33
|style="text-align:left" |chr_LRUH 2 chr_H 0 MitM_CTL NG maxT 100000 NG_n 5 run
|
|-
|style="text-align:center" |Terry Ligocki
|27,616,006
|27,552,018
|0.23%
|56.9
|0.31
|134.81
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 0 n 3 run
|
|-
|style="text-align:center" |Terry Ligocki
|27,552,018
|27,222,303
|1.2%
|92.3
|0.99
|82.96
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 6 n 2 run
|
|-
|style="text-align:center" |Terry Ligocki
|27,222,303
|26,626,978
|2.19%
|140.5
|1.18
|53.8
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 2 n 2 run
|
|-
|style="text-align:center" |Terry Ligocki
|26,626,978
|26,518,327
|0.41%
|126.9
|0.24
|58.31
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 3 run
|
|-
|style="text-align:center" |Terry Ligocki
|26,518,327
|26,334,644
|0.69%
|99.4
|0.51
|74.14
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 0 tH 2 n 4 run
|
|-
|style="text-align:center" |Terry Ligocki
|26,334,644
|26,076,261
|0.98%
|336.9
|0.21
|21.72
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 6 mod 6 n 1 run
|
|-
|style="text-align:center" |Terry Ligocki
|26,076,261
|25,828,854
|0.95%
|146.8
|0.47
|49.35
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 4 n 2 run
|
|-
|style="text-align:center" |Terry Ligocki
|25,828,854
|25,659,775
|0.65%
|203.4
|0.23
|35.28
|style="text-align:left" |chr_LRUH 8 chr_H 0 MitM_CTL NG maxT 30000 NG_n 6 run
|
|-
|style="text-align:center" |Terry Ligocki
|25,659,775
|25,532,914
|0.49%
|194.4
|0.18
|36.66
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 6 H 2 tH 2 n 5 run
|
|-
|style="text-align:center" |Terry Ligocki
|25,532,914
|25,318,355
|0.84%
|252.8
|0.24
|28.06
|style="text-align:left" |chr_LRUH 8 chr_H 6 MitM_CTL NG maxT 100000 NG_n 3 run
|
|-
|style="text-align:center" |Terry Ligocki
|25,318,355
|24,914,333
|1.6%
|446.1
|0.25
|15.77
|style="text-align:left" |chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 100000 NG_n 2 run
|
|-
|style="text-align:center" |Terry Ligocki
|24,914,333
|24,648,140
|1.07%
|258.6
|0.29
|26.76
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 16 H 2 tH 2 n 6 run
|
|-
|style="text-align:center" |Terry Ligocki
|24,648,140
|24,596,755
|0.21%
|322.7
|0.04
|21.21
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 16 mod 3 n 5 run
|
|-
|style="text-align:center" |Terry Ligocki
|24,596,755
|24,505,987
|0.37%
|292.0
|0.09
|23.4
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 6 H 1 tH 1 n 10 run
|
|-
|style="text-align:center" |Terry Ligocki
|24,505,987
|24,343,456
|0.66%
|1,067.8
|0.04
|6.37
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 100000 NG_n 6 run
|
|-
|style="text-align:center" |Terry Ligocki
|24,343,456
|24,116,020
|0.93%
|574.2
|0.11
|11.78
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 4 mod 2 n 4 run
|
|-
|style="text-align:center" |Terry Ligocki
|24,116,020
|24,008,284
|0.45%
|1,754.3
|0.02
|3.82
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 12 mod 1 n 4 run
|
|-
|style="text-align:center" |Terry Ligocki
|24,008,284
|23,502,214
|2.11%
|1,941.6
|0.07
|3.43
|style="text-align:left" |chr_LRUH 28 chr_LRUn 2 MitM_CTL NG maxT 100000 NG_n 8 run
|
|-
|style="text-align:center" |Terry Ligocki
|23,502,214
|23,332,229
|0.72%
|3,413.9
|0.01
|1.91
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 100000 NG_n 12 run
|
|-
|style="text-align:center" |Terry Ligocki
|23,332,229
|23,314,388
|0.08%
|1,182.1
|0.0
|5.48
|style="text-align:left" |chr_LRUH 9 chr_H 1 MitM_CTL NG maxT 100000 NG_n 13 run
|
|-
| style="text-align:center" |'''Stage 3 Cumulative'''
|'''28,189,617'''
|'''23,314,388'''
|'''17.3%'''
| '''12997.8'''
| ---
| ---
| style="text-align:left" | ---
| style="text-align:center" | ---
|-
|colspan="20" style="text-align:center"|'''Stage 4'''
|-
|style="text-align:center" |Andrew Ducharme
|23,314,388
|23,281,839
|0.14%
|2,227.8
|
|343.7
|style="text-align:left" |Enumerate.py with --block-multiple=1, max-loops=250_000, and --time=0.5.
|[https://discord.com/channels/960643023006490684/1369339127652159509/1421974799604912188 discord]
| rowspan="5" style="text-align:left"|Stage 4<br>
[https://drive.google.com/file/d/1N-vMMupgrsmIXe38sFczZRcF0t8XvgLT/view?usp=drive_link Holdouts]<br>
[https://drive.google.com/drive/folders/1FP7b5GHL9D-WQJ7E4XX3BnMBNuJUBmt1?usp=sharing Details]
|-
|style="text-align:center" |Andrew Ducharme
|23,281,839
|22,801,601
|2.06%
|~1,400
|
|
|style="text-align:left" |lr_enum_continue 10M steps
|[https://discord.com/channels/960643023006490684/1369339127652159509/1422298181693079743 discord]
|-
|style="text-align:center" |Andrew Ducharme
|22,801,601
|22,721,690
|0.35%
|3,334.6
|
|
|style="text-align:left" |Enumerate.py with --block-mult=3, --max-loops=250k, and --time=0.2.
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423019804985655418 discord]
|-
|style="text-align:center" |Andrew Ducharme
|22,721,690
|22,721,168
|0.002%
|3393.5
|
|
|style="text-align:left" |Enumerate.py with --block-mult=2, --max-loops=250k, and --time=0.2.
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 discord]
|-
| style="text-align:center" |'''Stage 4 Cumulative'''
|'''23,314,388'''
|'''22,721,168'''
|'''2.55%'''
|'''10,355.9'''
| ---
| ---
| ---
| ---
|-
|colspan="20" style="text-align:center"|'''Stage 5'''
|-
|style="text-align:center" |Terry Ligocki
|22,721,168
|20,405,295
|10.19%
|4,903.2
|0.13
|1.29
|
MitM_CTL RWL_mod sim 1001 maxT 100000 H 8 mod 3 n 12 run <br/>
MitM_CTL RWL_mod sim 1001 maxT 100000 H 6 mod 2 n 12 run <br/>
chr_LRUH 18 chr_H 16 MitM_CTL NG maxT 100000 NG_n 3 run <br/>
MitM_CTL CPS_LRU sim 1001 maxT 1000000 LRUH 20 H 2 tH 4 n 2 run <br/>
chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 1000000 NG_n 1 run <br/>
MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 4 n 4 run <br/>
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 9 n 1 run <br/>
MitM_CTL RWL_mod sim 1001 maxT 30000 H 12 mod 3 n 3 run <br/>
MitM_CTL RWL_mod sim 1001 maxT 300000 H 3 mod 2 n 2 run <br/>
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 1 tH 1 n 4 run <br/>
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 100000 NG_n 2 run <br/>
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 4 n 1 run <br/>
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 2 run <br/>
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 2 run <br/>
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 1 n 3 run <br/>
|
|rowspan="1" style="text-align:left"|Stage 5<br>
[https://drive.google.com/file/d/1jTyRvblSJnRDTwsfSCqD7kK0T4zxgU2A/view?usp=drive_link Holdouts]<br>
[https://drive.google.com/drive/folders/1APe-Vl8vcZOk4VMqRXHbPPxLanvKTmhA?usp=drive_link Details]
|-
| style="text-align:center" |'''Stage 5 Cumulative'''
|'''22,721,168'''
|'''20,405,295'''
|'''10.19%'''
|'''4903.2'''
| ---
| ---
| ---
| ---
|
|-
| colspan="20" style="text-align:center"|'''Stage 6'''
|-
|style="text-align:center"| Andrew Ducharme
|20,405,295
|20,387,509
|0.09%
|1797.2
|
|
|Enumerate.py with --block-mult=5, --max-loops=250k, --tape-limit=5000, and --time=0.3.
|[https://discord.com/channels/960643023006490684/1369339127652159509/1452727708479127633 discord]
| rowspan="8" |[https://drive.google.com/drive/folders/16uQS2oF136VO7-2znEjaACOWQiHHQFeX?usp=share_link Details]
|-
|style="text-align:center"| Andrew Ducharme
|20,387,509
|20,197,978
|0.93%
|6394.7
|
|
|Enumerate.py with --block-mult=20, --max-loops=1M, --tape-limit=5000, --max-steps-per-macro=100k, and --time=1.
|[https://discord.com/channels/960643023006490684/1369339127652159509/1459719965396566078 discord]
|-
|style="text-align:center"| Andrew Ducharme
|20,197,978
|19,879,953
|1.57%
|5748.1
|
|
|Enumerate.py with --block-mult=18, --max-loops=1M, --tape-limit=5000, --max-steps-per-macro=100k, and --time=1.
|[https://discord.com/channels/960643023006490684/1369339127652159509/1460718041577951365 discord]
|-
|style="text-align:center"| Andrew Ducharme
|19,879,953
|19,781,295
|0.50%
|6269.7
|
|
|Enumerate.py with --block-mult=21, --max-loops=1M, --tape-limit=5000, --max-steps-per-macro=100k, and --time=1.
|[https://discord.com/channels/960643023006490684/1369339127652159509/1461993117921054720 discord]
|-
|style="text-align:center"|Andrew Ducharme
|19,781,295
|19,303,801
|2.41%
|~2000
|
|
|FAR CPS_LRU maxT 1000000 LRUH 2 H 0 tH 0 n [1-10]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1464763753072427230 discord]
|-
|Andrew Ducharme
|19,303,801
|18,306,495
|5.17%
|~7000
|
|
|FAR CPS_LRU maxT 100000 LRUH 3 H 0 tH 0 n [1-31]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1469398962325684373 discord]
|-
|Andrew Ducharme
|18,306,495
|18,254,545
|0.28%
|???
|
|
|LRUH 1 H [0,1] tH [0,1] n [1-31]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1470510369687601375 discord]
|-
|Andrew Ducharme
|18,254,545
|18,195,192
|0.33%
|???
|
|
|maxT 100000 LRUH 2
|[https://discord.com/channels/960643023006490684/1369339127652159509/1472115313683202189 discord]
|-
| style="text-align:center" |'''Stage 6 Cumulative'''
|'''20,405,295'''
|'''18,195,192'''
|'''10.83%'''
|'''29209.7'''
| ---
| ---
| ---
| ---
|
|-
| style="text-align:center" |'''Overall Cumulative'''
|'''86,129,304'''
|'''18,195,192'''
|'''78.88%'''
|'''76419.8'''
| ---
| ---
| ---
| ---
|}
==References==
[[Category:BB Domains]][[Category:BB(7)]]
<references />

Graham's number

2026-03-14T08:15:06Z

C7X: /* History of Graham-beating TMs */

'''Graham's number''' (<math>g_{64}</math> or <math>G</math>) is a famously huge number which Martin Gardner claimed was the "largest number ever used in a serious mathematical proof" in 1977. Since it is one of the most famous large numbers, it has become a bit of a yardstick for measuring "hugeness". In the specific context of the Busy Beaver game, we can ask, what is the smallest <math>n</math> such that <math>BB(n) > g_{64}</math>. There is an active search for the smallest TM that runs for over Graham's number steps.

== Definition ==
See [https://en.wikipedia.org/wiki/Graham%27s_number Wikipedia article] for more detail
<math display="block">
\begin{array}{l}
g_0 & = & 4 \\
g_n & = & 3 \uparrow^{g_{n-1}} 3 & \text{if } n \ge 1 \\
\end{array}
</math>

Graham's number is <math>g_{64}</math>.

Using the [[fast-growing hierarchy]], <math>g_{64} < f_{\omega + 1}(64)</math>.

Let <math>N_G</math> be the smallest integer such that <math>S(N_G) > g_{64}</math>.

== Bounds ==
The current known bounds for <math>N_G</math> are:
<math display="block">
6 \le N_G \le 14
</math>

The lower bound comes from the proof that [[BB(5)]] = 47,176,870 and the upper bound from a specific 14-state TM found Racheline in August 2024 which runs for <math> > f_{\omega + 1}(65\,536) > g_{64} </math> steps.

=== History of Graham-beating TMs ===
There is no one authoritative source on the history of TMs beating Graham's number. Most were posted in personal blog posts, Googology pages/comments or on the bbchallenge Discord. They are often unverified and sometimes undocumented. This list is based upon historical accounts listed on [https://googology.fandom.com/wiki/Graham%27s_number#Comparison_with_busy_beaver_function Googologogy wiki], [https://cs.stackexchange.com/questions/69469/what-is-the-smallest-n-such-that-bbn-grahams-number/69476#69476 a 2017 CS Stack Exchange answer], [https://www.sligocki.com/2010/07/04/beating-grahams-number.html#results-from-the-future a 2022 history synthesis by Shawn Ligocki] and [https://bbchallenge.org/~pascal.michel/ha#topdown summary by Pascal Michel]. In general, these results are self-reported and we do not know of independent verification for most of them. Most of these were actually proven as bounds for the <math>\Sigma</math> function, but since <math>S(n) \ge \Sigma(n)</math> they apply to this formulation as well.
{| class="wikitable"
|+History of Graham-beating TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|91
|9 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|72
|13 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|64
|19 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/64-state-turing-machine-whose-output.html Blog Post] [https://web.archive.org/web/20130509222047/https://sites.google.com/site/res0001/surpassing-graham-s-number Summary]
|
|-
|25
|31 Mar 2013
|"Deedlit11"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines#A_New_Record!_Beating_Graham's_number_with_a_2-symbol_TM Googology Post]
|
|-
|24
|27 September 2013
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines?commentId=4400000000000011880 Googology Comment]
|
|-
|23
|7 October 2013
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines?commentId=4400000000000011880&replyId=4400000000000036301 Googology Comment]
|
|-
|22
|5 August 2014
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/NEWS!_I_found_a_22-state_machine_that_beats_G! Googology Post]
|
|-
|18
|24 Jul 2016
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham's_Number!?useskin=oasis Googology Post]
|
|-
|16
|26 Mar 2021
|Daniel Nagaj
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham%27s_Number!?commentId=4400000000000019187&replyId=4400000000000101283 Googology Comment] [http://morphett.info/turing/turing.html?197640ce0f380f8a6b0a4cdd138156a0 TM Definition]
|[https://www.sligocki.com/2022/07/11/bb-16-graham.html Analysis by Shawn Ligocki in 2022]
|-
|14
|17 Aug 2024
|[[User:Racheline|Racheline]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Discord Message]
|
|}

Graham's number

2026-03-14T07:52:55Z

C7X: /* History of Graham-beating TMs */

'''Graham's number''' (<math>g_{64}</math> or <math>G</math>) is a famously huge number which Martin Gardner claimed was the "largest number ever used in a serious mathematical proof" in 1977. Since it is one of the most famous large numbers, it has become a bit of a yardstick for measuring "hugeness". In the specific context of the Busy Beaver game, we can ask, what is the smallest <math>n</math> such that <math>BB(n) > g_{64}</math>. There is an active search for the smallest TM that runs for over Graham's number steps.

== Definition ==
See [https://en.wikipedia.org/wiki/Graham%27s_number Wikipedia article] for more detail
<math display="block">
\begin{array}{l}
g_0 & = & 4 \\
g_n & = & 3 \uparrow^{g_{n-1}} 3 & \text{if } n \ge 1 \\
\end{array}
</math>

Graham's number is <math>g_{64}</math>.

Using the [[fast-growing hierarchy]], <math>g_{64} < f_{\omega + 1}(64)</math>.

Let <math>N_G</math> be the smallest integer such that <math>S(N_G) > g_{64}</math>.

== Bounds ==
The current known bounds for <math>N_G</math> are:
<math display="block">
6 \le N_G \le 14
</math>

The lower bound comes from the proof that [[BB(5)]] = 47,176,870 and the upper bound from a specific 14-state TM found Racheline in August 2024 which runs for <math> > f_{\omega + 1}(65\,536) > g_{64} </math> steps.

=== History of Graham-beating TMs ===
There is no one authoritative source on the history of TMs beating Graham's number. Most were posted in personal blog posts, Googology pages/comments or on the bbchallenge Discord. They are often unverified and sometimes undocumented. This list is based upon historical accounts listed on [https://googology.fandom.com/wiki/Graham%27s_number#Comparison_with_busy_beaver_function Googologogy wiki], [https://cs.stackexchange.com/questions/69469/what-is-the-smallest-n-such-that-bbn-grahams-number/69476#69476 a 2017 CS Stack Exchange answer], [https://www.sligocki.com/2010/07/04/beating-grahams-number.html#results-from-the-future a 2022 history synthesis by Shawn Ligocki] and [https://bbchallenge.org/~pascal.michel/ha#topdown summary by Pascal Michel]. In general, these results are self-reported and we do not know of independent verification for most of them. Most of these were actually proven as bounds for the <math>\Sigma</math> function, but since <math>S(n) \ge \Sigma(n)</math> they apply to this formulation as well.
{| class="wikitable"
|+History of Graham-beating TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|91
|9 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|72
|13 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|64
|19 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/64-state-turing-machine-whose-output.html Blog Post] [https://web.archive.org/web/20130509222047/https://sites.google.com/site/res0001/surpassing-graham-s-number Summary]
|
|-
|25
|31 Mar 2013
|"Deedlit11"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines#A_New_Record!_Beating_Graham's_number_with_a_2-symbol_TM Googology Post]
|
|-
|24
|27 September 2013
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines?commentId=4400000000000011880 Googology Comment]
|
|-
|23
|7 October 2013
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines?commentId=4400000000000011880&replyId=4400000000000036301 Googology Comment]
|
|-
|18
|24 Jul 2016
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham's_Number!?useskin=oasis Googology Post]
|
|-
|16
|26 Mar 2021
|Daniel Nagaj
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham%27s_Number!?commentId=4400000000000019187&replyId=4400000000000101283 Googology Comment] [http://morphett.info/turing/turing.html?197640ce0f380f8a6b0a4cdd138156a0 TM Definition]
|[https://www.sligocki.com/2022/07/11/bb-16-graham.html Analysis by Shawn Ligocki in 2022]
|-
|14
|17 Aug 2024
|[[User:Racheline|Racheline]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Discord Message]
|
|}

Graham's number

2026-03-14T07:43:10Z

C7X: Improved to 18 on same day (for date proof, see the blog post's page history) /* History of Graham-beating TMs */

'''Graham's number''' (<math>g_{64}</math> or <math>G</math>) is a famously huge number which Martin Gardner claimed was the "largest number ever used in a serious mathematical proof" in 1977. Since it is one of the most famous large numbers, it has become a bit of a yardstick for measuring "hugeness". In the specific context of the Busy Beaver game, we can ask, what is the smallest <math>n</math> such that <math>BB(n) > g_{64}</math>. There is an active search for the smallest TM that runs for over Graham's number steps.

== Definition ==
See [https://en.wikipedia.org/wiki/Graham%27s_number Wikipedia article] for more detail
<math display="block">
\begin{array}{l}
g_0 & = & 4 \\
g_n & = & 3 \uparrow^{g_{n-1}} 3 & \text{if } n \ge 1 \\
\end{array}
</math>

Graham's number is <math>g_{64}</math>.

Using the [[fast-growing hierarchy]], <math>g_{64} < f_{\omega + 1}(64)</math>.

Let <math>N_G</math> be the smallest integer such that <math>S(N_G) > g_{64}</math>.

== Bounds ==
The current known bounds for <math>N_G</math> are:
<math display="block">
6 \le N_G \le 14
</math>

The lower bound comes from the proof that [[BB(5)]] = 47,176,870 and the upper bound from a specific 14-state TM found Racheline in August 2024 which runs for <math> > f_{\omega + 1}(65\,536) > g_{64} </math> steps.

=== History of Graham-beating TMs ===
There is no one authoritative source on the history of TMs beating Graham's number. Most were posted in personal blog posts, Googology pages/comments or on the bbchallenge Discord. They are often unverified and sometimes undocumented. This list is based upon historical accounts listed on [https://googology.fandom.com/wiki/Graham%27s_number#Comparison_with_busy_beaver_function Googologogy wiki], [https://cs.stackexchange.com/questions/69469/what-is-the-smallest-n-such-that-bbn-grahams-number/69476#69476 a 2017 CS Stack Exchange answer], [https://www.sligocki.com/2010/07/04/beating-grahams-number.html#results-from-the-future a 2022 history synthesis by Shawn Ligocki] and [https://bbchallenge.org/~pascal.michel/ha#topdown summary by Pascal Michel]. In general, these results are self-reported and we do not know of independent verification for most of them. Most of these were actually proven as bounds for the <math>\Sigma</math> function, but since <math>S(n) \ge \Sigma(n)</math> they apply to this formulation as well.
{| class="wikitable"
|+History of Graham-beating TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|91
|9 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|72
|13 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/small-turing-machine-whose-output.html Blog Post]
|
|-
|64
|19 Sep 2010
|"res001"
|[https://morethanazillion.blogspot.com/2010/09/64-state-turing-machine-whose-output.html Blog Post] [https://web.archive.org/web/20130509222047/https://sites.google.com/site/res0001/surpassing-graham-s-number Summary]
|
|-
|25
|31 Mar 2013
|"Deedlit11"
|[https://googology.fandom.com/wiki/User_blog:Deedlit11/Okay,_more_Turing_machines#A_New_Record!_Beating_Graham's_number_with_a_2-symbol_TM Googology Post]
|
|-
|18
|24 Jul 2016
|"Wythagoras"
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham's_Number!?useskin=oasis Googology Post]
|
|-
|16
|26 Mar 2021
|Daniel Nagaj
|[https://googology.fandom.com/wiki/User_blog:Wythagoras/The_nineteenth_Busy_Beaver_number_is_greater_than_Graham%27s_Number!?commentId=4400000000000019187&replyId=4400000000000101283 Googology Comment] [http://morphett.info/turing/turing.html?197640ce0f380f8a6b0a4cdd138156a0 TM Definition]
|[https://www.sligocki.com/2022/07/11/bb-16-graham.html Analysis by Shawn Ligocki in 2022]
|-
|14
|17 Aug 2024
|[[User:Racheline|Racheline]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Discord Message]
|
|}

Logical independence

2026-03-01T05:22:05Z

C7X: This is less invariant to how a theory is defined (as a set of axioms or as a deductively closed set of formulas), but equally general due to Craig's theorem

[[File:Independencechart.png|thumb|A chart showing which busy beaver numbers are independent of theories. Modification of a chart from a Vsauce video.]]
For any recursively enumerable and arithmetically sound axiomatic theory T, there exists an integer <math>N_T</math> such that T cannot prove the values of BB(n) for any <math>n \ge N_T</math>. For [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] (ZF), this value is known to be in the range:

<math display="block">6 \le N_{ZF} \le 432</math>

This lower bound comes from the fact that [[BB(5)]] has been proven in Rocq.<ref group="footnote">The fact that this theorem has been proven in Rocq technically does not imply that the theorem must be provable in ZF, because the consistency strength of Rocq is actually higher than that of ZF. However, the BB(5) proof does not use any techniques that could not be formalized within ZF.</ref> The upper bound comes from an explicit TM which enumerates all possible proofs in ZF and halts if it finds a proof 0 = 1. Assuming ZF is consistent and sound, then it cannot prove whether or not it is consistent, hence it cannot prove whether or not this specific TM halts.

Harvey Friedman spoke of embedding consistency statements within turing machines in a [https://fomarchive.ugent.be/2004-March/008003.html 2004 posting] on the "Foundations of Mathematics" mailing list. Scott Aaronson conjectured in his [[Busy Beaver Frontier]] survey that <math>N_{ZF} \le 20</math> and <math>N_{PA} \le 10</math>, where <math>PA</math> refers to the theory of [https://en.wikipedia.org/wiki/Peano%20axioms Peano Arithmetic]. Aaronson also mentioned how if BB(n) is independent of some formal theory T for a value n, the theory <math>PA</math> + “BB(n) = b” can prove the consistency of T, where b is the true value of BB(n).<ref name=":0">Scott Aaronson. 2020. [https://www.scottaaronson.com/papers/bb.pdf The Busy Beaver Frontier]. SIGACT News 51, 3 (August 2020), 32–54. https://doi.org/10.1145/3427361.3427369</ref>

== Axiom of Choice ==
Due to [[wikipedia:Absoluteness_(logic)#Shoenfield's_absoluteness_theorem|Shoenfield's absoluteness theorem]], it is known that any TM proven non-halting in ZFC can also be proven non-halting in ZF (and the converse is trivially true), therefore

<math display="block">N_{ZFC} = N_{ZF}</math>

Therefore we refer to ZF and <math>N_{ZF}</math> throughout this article since adding the Axiom of Choice does not have any effect on Turing machine decidability.

== History ==
There is no one authoritative source on the history of TMs independent of ZF, this is our best understanding of the history of TMs found. Mostly these are taken from Scott Aaronson's blog announcements and Busy Beaver Frontier or self-reported by the individuals who discovered them. The Aaronson-Yedida machine used a compiler called ''Laconic'', which was then updated with NQL or "Not Quite Laconic", while the current champion machines use a compiler built by Andrew J. Wade.

Note that these results also have not undergone formal verification, with the 7910 machine in particular relying on a result from Harvey Friedman that has no published proof.
{| class="wikitable"
|+History of ZF independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|7910
|May 2016
|Adam Yedidia and Scott Aaronson
|Yedidia and Aaronson 2016<ref>A. Yedidia and S. Aaronson. A relatively small Turing machine whose behavior is independent of set theory. Complex Systems, (25):4, 2016. https://arxiv.org/abs/1605.04343</ref>
|
|-
|748
|May 2016
|Stefan O’Rear
|[https://github.com/sorear/metamath-turing-machines/blob/master/zf2.nql Github NQL file], Busy Beaver Frontier<ref name=":0" />
|
|-
|745
|July 2023
|Johannes Riebel
|Riebel 2023 Bachelor Thesis<ref>Riebel, Johannes (March 2023). ''[https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf The Undecidability of BB(748): Understanding Gödel's Incompleteness Theorems]'' (PDF) (Bachelor's thesis). University of Augsburg.</ref>
|
|-
|643
|July 2024
|Rohan Ridenour
|[https://github.com/CatsAreFluffy/metamath-turing-machines Github NQL], [https://scottaaronson.blog/?p=8131 Aaronson Announcement]
|
|-
|636
|31 August 2024
|Rohan Ridenour
|[https://github.com/CatsAreFluffy/metamath-turing-machines Github NQL] ([https://github.com/CatsAreFluffy/metamath-turing-machines/commit/6fc33bef6ba8885d26aed94c83e88bdabbedb0f1 commit])
|
|-
|588
|12 July 2025
|Andrew J. Wade
|[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/30d2e3194866615f68dd2f5101cb300fb039adca commit])
|
|-
|549
|16 July 2025
|Andrew J. Wade
|[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/5d676aec074a94f598959cb3b7733a8f7781762f commit])
|
|-
|432
|19 Aug 2025
|Andrew J. Wade
|[https://codeberg.org/ajwade/turing_machine_explorer/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e git commit]
|
|}

Wade's 432-state champion also omits the [[wikipedia:Axiom_of_regularity|axiom of regularity]].

For Peano Arithmetic, the ZF independent machines are an upper bound. In general, a machine that is independent of a theory will also be independent of any theory with strictly lower [[wikipedia:Equiconsistency#Consistency_strength|consistency strength]]. For the theories in this article, Con(ZFC+Subtle)->Con(ZFC)->Con(PA).

{| class="wikitable"
|+History of Peano Arithmetic (PA) independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|372
|11 February 2026
|@LegionMammal978
|[https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py Github]
|
|}
This machine takes Wade's ZF champion and replaces the set theory axioms with one axiom schema of "adjunction + separation", which has been shown to interpret PA.<ref>https://mathoverflow.net/questions/508137/interpreting-pa-with-only-adjunction-separation/508161</ref>

=== Large cardinals ===
These tables are for theories stronger than ZFC.

{| class="wikitable"
|+History of ZFC + "There exist arbitrarily large subtle cardinals" independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|493
|25 February 2026
|@LegionMammal978
|[https://github.com/Saoi2/turing_machine_explorer/blob/main/subtle.tm Github]
|
|}
This machine takes Wade's ZF champion and adds the [[wikipedia:Axiom_of_choice|axiom of Choice]] and a statement from Harvey Friedman (Proposition 4.3 of [https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf "Primitive Independence Results"])<ref>H. Friedman, "[https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf Primitive Independence Results]" (2002). Accessed February 2026.</ref> which Friedman showed with ZFC to be equivalent to "There exist arbitrarily large subtle cardinals".

For reference, this theory is between the strength of "strongly unfoldable" and "<math>0^\sharp</math> exists" on the [[wikipedia:Large_cardinal|large cardinal hierarchy]] on Wikipedia.

Note: The initial machine produced in 2016 by Yedidia and Aaronson is designed to halt iff a certain statement created by Harvey Friedman is true.<ref name=":0" /> According to Friedman (without a published proof), this statement is independent of the theory "Stationary Ramsey Property" or ''SRP'', which is equiconsistent with the theory <math>ZFC + \{\text{``There is a }k\text{-subtle cardinal``}\mid k\in\N\}</math> which is also between the strength of "strongly unfoldable" and "<math>0^\sharp</math> exists" on the large cardinal hierarchy.<ref>tlonuqbar, "[https://mathoverflow.net/questions/508364/what-is-the-consistency-strength-of-srp-stationary-ramsey-property What is the consistency strength of SRP (Stationary Ramsey Property)?]" (2026). MathOverflow post, accessed February 2026.</ref> However, the theory used by the BB(493) machine is also independent of this theory.

== Footnotes ==
<references group="footnote"/>

== References ==
<references />

[[Category:Zoology]]

Talk:BB(11)

2026-02-28T02:43:17Z

C7X: Created page with "==Deletion== Should this page be deleted? There aren't pages for BB(9) and BB(10) ~~~~"

==Deletion==
Should this page be deleted? There aren't pages for BB(9) and BB(10) [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 02:43, 28 February 2026 (UTC)

Logical independence

2026-02-28T02:40:45Z

C7X: Add to citations /* Large cardinals */

[[File:Independencechart.png|thumb|A chart showing which busy beaver numbers are independent of theories. Modification of a chart from a Vsauce video.]]
For any computable and arithmetically sound axiomatic theory T, there exists an integer <math>N_T</math> such that T cannot prove the values of BB(n) for any <math>n \ge N_T</math>. For [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] (ZF), this value is known to be in the range:

<math display="block">6 \le N_{ZF} \le 432</math>

This lower bound comes from the fact that [[BB(5)]] has been proven in Rocq.<ref group="footnote">The fact that this theorem has been proven in Rocq technically does not imply that the theorem must be provable in ZF, because the consistency strength of Rocq is actually higher than that of ZF. However, the BB(5) proof does not use any techniques that could not be formalized within ZF.</ref> The upper bound comes from an explicit TM which enumerates all possible proofs in ZF and halts if it finds a proof 0 = 1. Assuming ZF is consistent and sound, then it cannot prove whether or not it is consistent, hence it cannot prove whether or not this specific TM halts.

Harvey Friedman spoke of embedding consistency statements within turing machines in a [https://fomarchive.ugent.be/2004-March/008003.html 2004 posting] on the "Foundations of Mathematics" mailing list. Scott Aaronson conjectured in his [[Busy Beaver Frontier]] survey that <math>N_{ZF} \le 20</math> and <math>N_{PA} \le 10</math>, where <math>PA</math> refers to the theory of [https://en.wikipedia.org/wiki/Peano%20axioms Peano Arithmetic]. Aaronson also mentioned how if BB(n) is independent of some formal theory T for a value n, the theory <math>PA</math> + “BB(n) = b” can prove the consistency of T, where b is the true value of BB(n).<ref name=":0">Scott Aaronson. 2020. [https://www.scottaaronson.com/papers/bb.pdf The Busy Beaver Frontier]. SIGACT News 51, 3 (August 2020), 32–54. https://doi.org/10.1145/3427361.3427369</ref>

== Axiom of Choice ==
Due to [[wikipedia:Absoluteness_(logic)#Shoenfield's_absoluteness_theorem|Shoenfield's absoluteness theorem]], it is known that any TM proven non-halting in ZFC can also be proven non-halting in ZF (and the converse is trivially true), therefore

<math display="block">N_{ZFC} = N_{ZF}</math>

Therefore we refer to ZF and <math>N_{ZF}</math> throughout this article since adding the Axiom of Choice does not have any effect on Turing machine decidability.

== History ==
There is no one authoritative source on the history of TMs independent of ZF, this is our best understanding of the history of TMs found. Mostly these are taken from Scott Aaronson's blog announcements and Busy Beaver Frontier or self-reported by the individuals who discovered them. The Aaronson-Yedida machine used a compiler called ''Laconic'', which was then updated with NQL or "Not Quite Laconic", while the current champion machines use a compiler built by Andrew J. Wade.

Note that these results also have not undergone formal verification, with the 7910 machine in particular relying on a result from Harvey Friedman that has no published proof.
{| class="wikitable"
|+History of ZF independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|7910
|May 2016
|Adam Yedidia and Scott Aaronson
|Yedidia and Aaronson 2016<ref>A. Yedidia and S. Aaronson. A relatively small Turing machine whose behavior is independent of set theory. Complex Systems, (25):4, 2016. https://arxiv.org/abs/1605.04343</ref>
|
|-
|748
|May 2016
|Stefan O’Rear
|[https://github.com/sorear/metamath-turing-machines/blob/master/zf2.nql Github NQL file], Busy Beaver Frontier<ref name=":0" />
|
|-
|745
|July 2023
|Johannes Riebel
|Riebel 2023 Bachelor Thesis<ref>Riebel, Johannes (March 2023). ''[https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf The Undecidability of BB(748): Understanding Gödel's Incompleteness Theorems]'' (PDF) (Bachelor's thesis). University of Augsburg.</ref>
|
|-
|643
|July 2024
|Rohan Ridenour
|[https://github.com/CatsAreFluffy/metamath-turing-machines Github NQL], [https://scottaaronson.blog/?p=8131 Aaronson Announcement]
|
|-
|636
|31 August 2024
|Rohan Ridenour
|[https://github.com/CatsAreFluffy/metamath-turing-machines Github NQL] ([https://github.com/CatsAreFluffy/metamath-turing-machines/commit/6fc33bef6ba8885d26aed94c83e88bdabbedb0f1 commit])
|
|-
|588
|12 July 2025
|Andrew J. Wade
|[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/30d2e3194866615f68dd2f5101cb300fb039adca commit])
|
|-
|549
|16 July 2025
|Andrew J. Wade
|[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/5d676aec074a94f598959cb3b7733a8f7781762f commit])
|
|-
|432
|19 Aug 2025
|Andrew J. Wade
|[https://codeberg.org/ajwade/turing_machine_explorer/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e git commit]
|
|}

Wade's 432-state champion also omits the [[wikipedia:Axiom_of_regularity|axiom of regularity]].

For Peano Arithmetic, the ZF independent machines are an upper bound. In general, a machine that is independent of a theory will also be independent of any theory with strictly lower [[wikipedia:Equiconsistency#Consistency_strength|consistency strength]]. For the theories in this article, Con(ZFC+Subtle)->Con(ZFC)->Con(PA).

{| class="wikitable"
|+History of Peano Arithmetic (PA) independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|372
|11 February 2026
|@LegionMammal978
|[https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py Github]
|
|}
This machine takes Wade's ZF champion and replaces the set theory axioms with one axiom schema of "adjunction + separation", which has been shown to interpret PA.<ref>https://mathoverflow.net/questions/508137/interpreting-pa-with-only-adjunction-separation/508161</ref>

=== Large cardinals ===
These tables are for theories stronger than ZFC.

{| class="wikitable"
|+History of ZFC + "There exist arbitrarily large subtle cardinals" independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|493
|25 February 2026
|@LegionMammal978
|[https://github.com/Saoi2/turing_machine_explorer/blob/main/subtle.tm Github]
|
|}
This machine takes Wade's ZF champion and adds the [[wikipedia:Axiom_of_choice|axiom of Choice]] and a statement from Harvey Friedman (Proposition 4.3 of [https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf "Primitive Independence Results"])<ref>H. Friedman, "[https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf Primitive Independence Results]" (2002). Accessed February 2026.</ref> which Friedman showed with ZFC to be equivalent to "There exist arbitrarily large subtle cardinals".

For reference, this theory is between the strength of "strongly unfoldable" and "<math>0^\sharp</math> exists" on the [[wikipedia:Large_cardinal|large cardinal hierarchy]] on Wikipedia.

Note: The initial machine produced in 2016 by Yedidia and Aaronson is designed to halt iff a certain statement created by Harvey Friedman is true.<ref name=":0" /> According to Friedman (without a published proof), this statement is independent of the theory "Stationary Ramsey Property" or ''SRP'', which is equiconsistent with the theory <math>ZFC + \{\text{``There is a }k\text{-subtle cardinal``}\mid k\in\N\}</math> which is also between the strength of "strongly unfoldable" and "<math>0^\sharp</math> exists" on the large cardinal hierarchy.<ref>tlonuqbar, "[https://mathoverflow.net/questions/508364/what-is-the-consistency-strength-of-srp-stationary-ramsey-property What is the consistency strength of SRP (Stationary Ramsey Property)?]" (2026). MathOverflow post, accessed February 2026.</ref> However, the theory used by the BB(493) machine is also independent of this theory.

== Footnotes ==
<references group="footnote"/>

== References ==
<references />

[[Category:Zoology]]

Logical independence

2026-02-28T02:36:29Z

C7X: MathJax

[[File:Independencechart.png|thumb|A chart showing which busy beaver numbers are independent of theories. Modification of a chart from a Vsauce video.]]
For any computable and arithmetically sound axiomatic theory T, there exists an integer <math>N_T</math> such that T cannot prove the values of BB(n) for any <math>n \ge N_T</math>. For [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] (ZF), this value is known to be in the range:

<math display="block">6 \le N_{ZF} \le 432</math>

This lower bound comes from the fact that [[BB(5)]] has been proven in Rocq.<ref group="footnote">The fact that this theorem has been proven in Rocq technically does not imply that the theorem must be provable in ZF, because the consistency strength of Rocq is actually higher than that of ZF. However, the BB(5) proof does not use any techniques that could not be formalized within ZF.</ref> The upper bound comes from an explicit TM which enumerates all possible proofs in ZF and halts if it finds a proof 0 = 1. Assuming ZF is consistent and sound, then it cannot prove whether or not it is consistent, hence it cannot prove whether or not this specific TM halts.

Harvey Friedman spoke of embedding consistency statements within turing machines in a [https://fomarchive.ugent.be/2004-March/008003.html 2004 posting] on the "Foundations of Mathematics" mailing list. Scott Aaronson conjectured in his [[Busy Beaver Frontier]] survey that <math>N_{ZF} \le 20</math> and <math>N_{PA} \le 10</math>, where <math>PA</math> refers to the theory of [https://en.wikipedia.org/wiki/Peano%20axioms Peano Arithmetic]. Aaronson also mentioned how if BB(n) is independent of some formal theory T for a value n, the theory <math>PA</math> + “BB(n) = b” can prove the consistency of T, where b is the true value of BB(n).<ref name=":0">Scott Aaronson. 2020. [https://www.scottaaronson.com/papers/bb.pdf The Busy Beaver Frontier]. SIGACT News 51, 3 (August 2020), 32–54. https://doi.org/10.1145/3427361.3427369</ref>

== Axiom of Choice ==
Due to [[wikipedia:Absoluteness_(logic)#Shoenfield's_absoluteness_theorem|Shoenfield's absoluteness theorem]], it is known that any TM proven non-halting in ZFC can also be proven non-halting in ZF (and the converse is trivially true), therefore

<math display="block">N_{ZFC} = N_{ZF}</math>

Therefore we refer to ZF and <math>N_{ZF}</math> throughout this article since adding the Axiom of Choice does not have any effect on Turing machine decidability.

== History ==
There is no one authoritative source on the history of TMs independent of ZF, this is our best understanding of the history of TMs found. Mostly these are taken from Scott Aaronson's blog announcements and Busy Beaver Frontier or self-reported by the individuals who discovered them. The Aaronson-Yedida machine used a compiler called ''Laconic'', which was then updated with NQL or "Not Quite Laconic", while the current champion machines use a compiler built by Andrew J. Wade.

Note that these results also have not undergone formal verification, with the 7910 machine in particular relying on a result from Harvey Friedman that has no published proof.
{| class="wikitable"
|+History of ZF independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|7910
|May 2016
|Adam Yedidia and Scott Aaronson
|Yedidia and Aaronson 2016<ref>A. Yedidia and S. Aaronson. A relatively small Turing machine whose behavior is independent of set theory. Complex Systems, (25):4, 2016. https://arxiv.org/abs/1605.04343</ref>
|
|-
|748
|May 2016
|Stefan O’Rear
|[https://github.com/sorear/metamath-turing-machines/blob/master/zf2.nql Github NQL file], Busy Beaver Frontier<ref name=":0" />
|
|-
|745
|July 2023
|Johannes Riebel
|Riebel 2023 Bachelor Thesis<ref>Riebel, Johannes (March 2023). ''[https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf The Undecidability of BB(748): Understanding Gödel's Incompleteness Theorems]'' (PDF) (Bachelor's thesis). University of Augsburg.</ref>
|
|-
|643
|July 2024
|Rohan Ridenour
|[https://github.com/CatsAreFluffy/metamath-turing-machines Github NQL], [https://scottaaronson.blog/?p=8131 Aaronson Announcement]
|
|-
|636
|31 August 2024
|Rohan Ridenour
|[https://github.com/CatsAreFluffy/metamath-turing-machines Github NQL] ([https://github.com/CatsAreFluffy/metamath-turing-machines/commit/6fc33bef6ba8885d26aed94c83e88bdabbedb0f1 commit])
|
|-
|588
|12 July 2025
|Andrew J. Wade
|[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/30d2e3194866615f68dd2f5101cb300fb039adca commit])
|
|-
|549
|16 July 2025
|Andrew J. Wade
|[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/5d676aec074a94f598959cb3b7733a8f7781762f commit])
|
|-
|432
|19 Aug 2025
|Andrew J. Wade
|[https://codeberg.org/ajwade/turing_machine_explorer/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e git commit]
|
|}

Wade's 432-state champion also omits the [[wikipedia:Axiom_of_regularity|axiom of regularity]].

For Peano Arithmetic, the ZF independent machines are an upper bound. In general, a machine that is independent of a theory will also be independent of any theory with strictly lower [[wikipedia:Equiconsistency#Consistency_strength|consistency strength]]. For the theories in this article, Con(ZFC+Subtle)->Con(ZFC)->Con(PA).

{| class="wikitable"
|+History of Peano Arithmetic (PA) independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|372
|11 February 2026
|@LegionMammal978
|[https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py Github]
|
|}
This machine takes Wade's ZF champion and replaces the set theory axioms with one axiom schema of "adjunction + separation", which has been shown to interpret PA.<ref>https://mathoverflow.net/questions/508137/interpreting-pa-with-only-adjunction-separation/508161</ref>

=== Large cardinals ===
These tables are for theories stronger than ZFC.

{| class="wikitable"
|+History of ZFC + "There exist arbitrarily large subtle cardinals" independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|493
|25 February 2026
|@LegionMammal978
|[https://github.com/Saoi2/turing_machine_explorer/blob/main/subtle.tm Github]
|
|}
This machine takes Wade's ZF champion and adds the [[wikipedia:Axiom_of_choice|axiom of Choice]] and a statement from Harvey Friedman (Proposition 4.3 of [https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf "Primitive Independence Results"])<ref>https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf</ref> which Friedman showed with ZFC to be equivalent to "There exist arbitrarily large subtle cardinals".

For reference, this theory is between the strength of "strongly unfoldable" and "<math>0^\sharp</math> exists" on the [[wikipedia:Large_cardinal|large cardinal hierarchy]] on Wikipedia.

Note: The initial machine produced in 2016 by Yedidia and Aaronson is designed to halt iff a certain statement created by Harvey Friedman is true.<ref name=":0" /> According to Friedman (without a published proof), this statement is independent of the theory "Stationary Ramsey Property" or ''SRP'', which is equiconsistent with the theory <math>ZFC + \{\text{``There is a }k\text{-subtle cardinal``}\mid k\in\N\}</math> which is also between the strength of "strongly unfoldable" and "<math>0^\sharp</math> exists" on the large cardinal hierarchy.<ref>https://mathoverflow.net/questions/508364/what-is-the-consistency-strength-of-srp-stationary-ramsey-property</ref> However, the theory used by the BB(493) machine is also independent of this theory.

== Footnotes ==
<references group="footnote"/>

== References ==
<references />

[[Category:Zoology]]

Logical independence

2026-02-28T02:34:27Z

C7X: Infinitely many axioms

[[File:Independencechart.png|thumb|A chart showing which busy beaver numbers are independent of theories. Modification of a chart from a Vsauce video.]]
For any computable and arithmetically sound axiomatic theory T, there exists an integer <math>N_T</math> such that T cannot prove the values of BB(n) for any <math>n \ge N_T</math>. For [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] (ZF), this value is known to be in the range:

<math display="block">6 \le N_{ZF} \le 432</math>

This lower bound comes from the fact that [[BB(5)]] has been proven in Rocq.<ref group="footnote">The fact that this theorem has been proven in Rocq technically does not imply that the theorem must be provable in ZF, because the consistency strength of Rocq is actually higher than that of ZF. However, the BB(5) proof does not use any techniques that could not be formalized within ZF.</ref> The upper bound comes from an explicit TM which enumerates all possible proofs in ZF and halts if it finds a proof 0 = 1. Assuming ZF is consistent and sound, then it cannot prove whether or not it is consistent, hence it cannot prove whether or not this specific TM halts.

Harvey Friedman spoke of embedding consistency statements within turing machines in a [https://fomarchive.ugent.be/2004-March/008003.html 2004 posting] on the "Foundations of Mathematics" mailing list. Scott Aaronson conjectured in his [[Busy Beaver Frontier]] survey that <math>N_{ZF} \le 20</math> and <math>N_{PA} \le 10</math>, where <math>PA</math> refers to the theory of [https://en.wikipedia.org/wiki/Peano%20axioms Peano Arithmetic]. Aaronson also mentioned how if BB(n) is independent of some formal theory T for a value n, the theory <math>PA</math> + “BB(n) = b” can prove the consistency of T, where b is the true value of BB(n).<ref name=":0">Scott Aaronson. 2020. [https://www.scottaaronson.com/papers/bb.pdf The Busy Beaver Frontier]. SIGACT News 51, 3 (August 2020), 32–54. https://doi.org/10.1145/3427361.3427369</ref>

== Axiom of Choice ==
Due to [[wikipedia:Absoluteness_(logic)#Shoenfield's_absoluteness_theorem|Shoenfield's absoluteness theorem]], it is known that any TM proven non-halting in ZFC can also be proven non-halting in ZF (and the converse is trivially true), therefore

<math display="block">N_{ZFC} = N_{ZF}</math>

Therefore we refer to ZF and <math>N_{ZF}</math> throughout this article since adding the Axiom of Choice does not have any effect on Turing machine decidability.

== History ==
There is no one authoritative source on the history of TMs independent of ZF, this is our best understanding of the history of TMs found. Mostly these are taken from Scott Aaronson's blog announcements and Busy Beaver Frontier or self-reported by the individuals who discovered them. The Aaronson-Yedida machine used a compiler called ''Laconic'', which was then updated with NQL or "Not Quite Laconic", while the current champion machines use a compiler built by Andrew J. Wade.

Note that these results also have not undergone formal verification, with the 7910 machine in particular relying on a result from Harvey Friedman that has no published proof.
{| class="wikitable"
|+History of ZF independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|7910
|May 2016
|Adam Yedidia and Scott Aaronson
|Yedidia and Aaronson 2016<ref>A. Yedidia and S. Aaronson. A relatively small Turing machine whose behavior is independent of set theory. Complex Systems, (25):4, 2016. https://arxiv.org/abs/1605.04343</ref>
|
|-
|748
|May 2016
|Stefan O’Rear
|[https://github.com/sorear/metamath-turing-machines/blob/master/zf2.nql Github NQL file], Busy Beaver Frontier<ref name=":0" />
|
|-
|745
|July 2023
|Johannes Riebel
|Riebel 2023 Bachelor Thesis<ref>Riebel, Johannes (March 2023). ''[https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf The Undecidability of BB(748): Understanding Gödel's Incompleteness Theorems]'' (PDF) (Bachelor's thesis). University of Augsburg.</ref>
|
|-
|643
|July 2024
|Rohan Ridenour
|[https://github.com/CatsAreFluffy/metamath-turing-machines Github NQL], [https://scottaaronson.blog/?p=8131 Aaronson Announcement]
|
|-
|636
|31 August 2024
|Rohan Ridenour
|[https://github.com/CatsAreFluffy/metamath-turing-machines Github NQL] ([https://github.com/CatsAreFluffy/metamath-turing-machines/commit/6fc33bef6ba8885d26aed94c83e88bdabbedb0f1 commit])
|
|-
|588
|12 July 2025
|Andrew J. Wade
|[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/30d2e3194866615f68dd2f5101cb300fb039adca commit])
|
|-
|549
|16 July 2025
|Andrew J. Wade
|[https://github.com/andrew-j-wade/metamath-turing-machines Github NQL] ([https://github.com/andrew-j-wade/metamath-turing-machines/commit/5d676aec074a94f598959cb3b7733a8f7781762f commit])
|
|-
|432
|19 Aug 2025
|Andrew J. Wade
|[https://codeberg.org/ajwade/turing_machine_explorer/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e git commit]
|
|}

Wade's 432-state champion also omits the [[wikipedia:Axiom_of_regularity|axiom of regularity]].

For Peano Arithmetic, the ZF independent machines are an upper bound. In general, a machine that is independent of a theory will also be independent of any theory with strictly lower [[wikipedia:Equiconsistency#Consistency_strength|consistency strength]]. For the theories in this article, Con(ZFC+Subtle)->Con(ZFC)->Con(PA).

{| class="wikitable"
|+History of Peano Arithmetic (PA) independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|372
|11 February 2026
|@LegionMammal978
|[https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py Github]
|
|}
This machine takes Wade's ZF champion and replaces the set theory axioms with one axiom schema of "adjunction + separation", which has been shown to interpret PA.<ref>https://mathoverflow.net/questions/508137/interpreting-pa-with-only-adjunction-separation/508161</ref>

=== Large cardinals ===
These tables are for theories stronger than ZFC.

{| class="wikitable"
|+History of ZFC + "There exist arbitrarily large subtle cardinals" independent TMs
!States
!Date
!Discoverer
!Source
!Verification
|-
|493
|25 February 2026
|@LegionMammal978
|[https://github.com/Saoi2/turing_machine_explorer/blob/main/subtle.tm Github]
|
|}
This machine takes Wade's ZF champion and adds the [[wikipedia:Axiom_of_choice|axiom of Choice]] and a statement from Harvey Friedman (Proposition 4.3 of [https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf "Primitive Independence Results"])<ref>https://bpb-us-w2.wpmucdn.com/u.osu.edu/dist/1/1952/files/2014/01/PrimitiveIndResults071302-189vmn0.pdf</ref> which Friedman showed with ZFC to be equivalent to "There exist arbitrarily large subtle cardinals".

For reference, this theory is between the strength of "strongly unfoldable" and "0# exists" on the [[wikipedia:Large_cardinal|large cardinal hierarchy]] on Wikipedia.

Note: The initial machine produced in 2016 by Yedidia and Aaronson is designed to halt iff a certain statement created by Harvey Friedman is true.<ref name=":0" /> According to Friedman (without a published proof), this statement is independent of the theory "Stationary Ramsey Property" or ''SRP'', which is equiconsistent with the theory <math>ZFC + \{\text{``There is a k-subtle cardinal``}\mid k\in\N\}</math> which is also between the strength of "strongly unfoldable" and "0# exists" on the large cardinal hierarchy.<ref>https://mathoverflow.net/questions/508364/what-is-the-consistency-strength-of-srp-stationary-ramsey-property</ref> However, the theory used by the BB(493) machine is also independent of this theory.

== Footnotes ==
<references group="footnote"/>

== References ==
<references />

[[Category:Zoology]]

Talk:Logical independence

2025-11-06T07:59:45Z

C7X: /* Extensionality */

A good introduction can come from here: https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf,

== ZF vs. ZFC ==

The way this article is written makes it unclear what specifically applies for ZF vs. ZFC. I don't know the specifics myself, so some clarification would be nice. [[User:XnoobSpeakable|XnoobSpeakable]]

: Currently, all of the machines listed in this article except one are written to start enumerating theorems of ZF-Regularity and halt if they find a contradiction, so they halt iff Con(ZF-Regularity) is false. Con(ZF-Regularity) is equivalent to Con(ZF) although I don't know how to prove that, but I do know how to prove that Con(ZF) is equivalent to Con(ZFC): if you assume ZF is consistent, there is a model of it, then you take the constructible universe of that model to obtain a model of ZFC, so then ZFC is consistent. The one different machine is the original Aaronson-Yedidia machine. Instead it uses one of Friedman's statements, which is independent of both ZF and ZFC (and even ZFC+some large cardinals!) [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 20:04, 21 July 2025 (UTC)

:: Interesting, I didn't realize these were for ZF minus Axiom of Regularity. Looks like Wikipedia has some sources for Axiom of Regularity being "relatively consistent" with the rest of ZF: https://en.wikipedia.org/wiki/Axiom_of_regularity#Regularity_and_the_rest_of_ZF(C)_axioms although I haven't tried to read any of them. I suppose the motivation to remove regularity is just that it makes the TMs smaller since they have fewer axioms to enumerate? Does it sound correct to say that <math>N_{ZF} = N_{ZFC} = N_{ZF-Regularity}</math>? I guess maybe that is not known, but all the current TMs halt iff Con(ZF-Regularity) is false, so then are upper bounds for all three of these numbers? [[User:Sligocki|Sligocki]] ([[User talk:Sligocki|talk]]) 21:24, 21 July 2025 (UTC)

::: I can say <math>N_{ZF} = N_{ZFC}</math> with confidence, since the statement that a TM runs forever is <math>\Pi^0_1</math> respectively, and by [https://en.wikipedia.org/wiki/Absoluteness_(logic)#Shoenfield's_absoluteness_theorem Shoenfield's absoluteness theorem] ZF proves a <math>\Pi^0_1</math> statement iff ZFC proves it. (This theorem is proven via pretty much the same trick that I mentioned above, passing to the constructible universe of a given model of ZF to obtain choice.) For ZF without regularity, the standard machinery used in work there seems to be working with [https://en.wikipedia.org/wiki/Permutation_model permutation models] (also see [https://en.wikipedia.org/wiki/Axiom_of_regularity#Regularity_and_the_rest_of_ZF(C)_axioms]), but I don't know how to work with those so I can't say anything about <math>N_{ZF-Regularity}</math> unfortunately [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 00:06, 22 July 2025 (UTC)

== Extensionality ==
Would removing the axiom of extensionality from the machines be helpful for shrinking them further?

Z minus extensionality is equiconsistent with Z, but ZF minus extensionality is also equiconsistent with Z. (R. O. Gandy, "[https://doi.org/10.2307/2271630 On the Axiom of Extensionality]") Both of these theories still have regularity though, so it would still have to be checked that this result holds when replacement is removed. Gandy says that the reason replacement becomes weak without extensionality is because y = z occurs in the statement of replacement, so maybe using collection instead could get its consistency strength back up to ZFC's. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 07:58, 6 November 2025 (UTC)

Talk:Logical independence

2025-11-06T07:59:16Z

C7X: /* Extensionality */

A good introduction can come from here: https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf,

== ZF vs. ZFC ==

The way this article is written makes it unclear what specifically applies for ZF vs. ZFC. I don't know the specifics myself, so some clarification would be nice. [[User:XnoobSpeakable|XnoobSpeakable]]

: Currently, all of the machines listed in this article except one are written to start enumerating theorems of ZF-Regularity and halt if they find a contradiction, so they halt iff Con(ZF-Regularity) is false. Con(ZF-Regularity) is equivalent to Con(ZF) although I don't know how to prove that, but I do know how to prove that Con(ZF) is equivalent to Con(ZFC): if you assume ZF is consistent, there is a model of it, then you take the constructible universe of that model to obtain a model of ZFC, so then ZFC is consistent. The one different machine is the original Aaronson-Yedidia machine. Instead it uses one of Friedman's statements, which is independent of both ZF and ZFC (and even ZFC+some large cardinals!) [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 20:04, 21 July 2025 (UTC)

:: Interesting, I didn't realize these were for ZF minus Axiom of Regularity. Looks like Wikipedia has some sources for Axiom of Regularity being "relatively consistent" with the rest of ZF: https://en.wikipedia.org/wiki/Axiom_of_regularity#Regularity_and_the_rest_of_ZF(C)_axioms although I haven't tried to read any of them. I suppose the motivation to remove regularity is just that it makes the TMs smaller since they have fewer axioms to enumerate? Does it sound correct to say that <math>N_{ZF} = N_{ZFC} = N_{ZF-Regularity}</math>? I guess maybe that is not known, but all the current TMs halt iff Con(ZF-Regularity) is false, so then are upper bounds for all three of these numbers? [[User:Sligocki|Sligocki]] ([[User talk:Sligocki|talk]]) 21:24, 21 July 2025 (UTC)

::: I can say <math>N_{ZF} = N_{ZFC}</math> with confidence, since the statement that a TM runs forever is <math>\Pi^0_1</math> respectively, and by [https://en.wikipedia.org/wiki/Absoluteness_(logic)#Shoenfield's_absoluteness_theorem Shoenfield's absoluteness theorem] ZF proves a <math>\Pi^0_1</math> statement iff ZFC proves it. (This theorem is proven via pretty much the same trick that I mentioned above, passing to the constructible universe of a given model of ZF to obtain choice.) For ZF without regularity, the standard machinery used in work there seems to be working with [https://en.wikipedia.org/wiki/Permutation_model permutation models] (also see [https://en.wikipedia.org/wiki/Axiom_of_regularity#Regularity_and_the_rest_of_ZF(C)_axioms]), but I don't know how to work with those so I can't say anything about <math>N_{ZF-Regularity}</math> unfortunately [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 00:06, 22 July 2025 (UTC)

== Extensionality ==
Would removing the axiom of extensionality from the machines be helpful for shrinking them further?

Z minus extensionality is equiconsistent with Z, but ZF minus extensionality is also equiconsistent with Z. (R. O. Gandy, "[https://doi.org/10.2307/2271630 On the Axiom of Extensionality]") Both of these theories still have regularity though, so it would still have to be checked that this result holds when replacement is removed. Gandy says that the reason replacement becomes weak is because y = z becomes hard to prove without extensionality, so maybe using collection instead could get its consistency strength back up to ZFC's. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 07:58, 6 November 2025 (UTC)

Talk:Logical independence

2025-11-06T07:58:53Z

C7X: /* ZF vs. ZFC */

A good introduction can come from here: https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf,

== ZF vs. ZFC ==

The way this article is written makes it unclear what specifically applies for ZF vs. ZFC. I don't know the specifics myself, so some clarification would be nice. [[User:XnoobSpeakable|XnoobSpeakable]]

: Currently, all of the machines listed in this article except one are written to start enumerating theorems of ZF-Regularity and halt if they find a contradiction, so they halt iff Con(ZF-Regularity) is false. Con(ZF-Regularity) is equivalent to Con(ZF) although I don't know how to prove that, but I do know how to prove that Con(ZF) is equivalent to Con(ZFC): if you assume ZF is consistent, there is a model of it, then you take the constructible universe of that model to obtain a model of ZFC, so then ZFC is consistent. The one different machine is the original Aaronson-Yedidia machine. Instead it uses one of Friedman's statements, which is independent of both ZF and ZFC (and even ZFC+some large cardinals!) [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 20:04, 21 July 2025 (UTC)

:: Interesting, I didn't realize these were for ZF minus Axiom of Regularity. Looks like Wikipedia has some sources for Axiom of Regularity being "relatively consistent" with the rest of ZF: https://en.wikipedia.org/wiki/Axiom_of_regularity#Regularity_and_the_rest_of_ZF(C)_axioms although I haven't tried to read any of them. I suppose the motivation to remove regularity is just that it makes the TMs smaller since they have fewer axioms to enumerate? Does it sound correct to say that <math>N_{ZF} = N_{ZFC} = N_{ZF-Regularity}</math>? I guess maybe that is not known, but all the current TMs halt iff Con(ZF-Regularity) is false, so then are upper bounds for all three of these numbers? [[User:Sligocki|Sligocki]] ([[User talk:Sligocki|talk]]) 21:24, 21 July 2025 (UTC)

::: I can say <math>N_{ZF} = N_{ZFC}</math> with confidence, since the statement that a TM runs forever is <math>\Pi^0_1</math> respectively, and by [https://en.wikipedia.org/wiki/Absoluteness_(logic)#Shoenfield's_absoluteness_theorem Shoenfield's absoluteness theorem] ZF proves a <math>\Pi^0_1</math> statement iff ZFC proves it. (This theorem is proven via pretty much the same trick that I mentioned above, passing to the constructible universe of a given model of ZF to obtain choice.) For ZF without regularity, the standard machinery used in work there seems to be working with [https://en.wikipedia.org/wiki/Permutation_model permutation models] (also see [https://en.wikipedia.org/wiki/Axiom_of_regularity#Regularity_and_the_rest_of_ZF(C)_axioms]), but I don't know how to work with those so I can't say anything about <math>N_{ZF-Regularity}</math> unfortunately [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 00:06, 22 July 2025 (UTC)

== Extensionality ==
Could there be future progress made by removing the axiom of extensionality from the machines?

Z minus extensionality is equiconsistent with Z, but ZF minus extensionality is also equiconsistent with Z. (R. O. Gandy, "[https://doi.org/10.2307/2271630 On the Axiom of Extensionality]") Both of these theories still have regularity though, so it would still have to be checked that this result holds when replacement is removed. Gandy says that the reason replacement becomes weak is because y = z becomes hard to prove without extensionality, so maybe using collection instead could get its consistency strength back up to ZFC's. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 07:58, 6 November 2025 (UTC)

1RB0LC 1RC1RA 1RD0RF 0LE--- 1LA1LE 0RA1RF

2025-10-12T18:26:24Z

C7X:

{{Stub}}

{{TM|1RB0LC_1RC1RA_1RD0RF_0LE---_1LA1LE_0RA1RF}} hasn't yet been analyzed.

[[File:30kstepsmachine.jpg|thumb|The machine after 30,000 steps]]
[[Category:Stubs]]

Cryptids

2025-09-03T15:27:36Z

C7X: State count improvement for ZF /* Larger Cryptids */

[[File:Lovecraft beaver.png|alt=A monstrous beaver in the style of HP Lovecraft with pink tentacles coming out of its mouth, 5 red spider eyes, horns on its head, elbows and tail, moss colored fur, sharp purple claws and webbed feet.|thumb|Lovecraftian Beaver fan art made by Lauren]]
'''Cryptids''' are Turing Machines whose behavior (when started on a blank tape) can be described completely by a relatively simple mathematical rule, but where that rule falls into a class of unsolved (and presumed hard) mathematical problems. This definition is somewhat subjective (What counts as a simple rule? What counts as a hard problem?). In practice, most currently known small Cryptids have [[Collatz-like]] behavior. In other words, the halting problem from blank tape of Cryptids is mathematically-hard.

If there exists a Cryptid with n states and m symbols, then BB(n, m) cannot be solved without solving this hard math problem.

The name Cryptid was proposed by Shawn Ligocki in an Oct 2023 [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html blog post] announcing the discovery of [[Bigfoot]].

== Cryptids at the Edge ==

This is a list of notable Minimal Cryptids (Cryptids in a [[:Category:BB_Domains|domain]] with no strictly smaller known Cryptid). All of these Cryptids were "discovered in the wild" rather than "constructed".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|
|-
|[[Hydra]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|
|-
|[[Bonus cryptid]]
|[[BB(2,5)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|June 2024
|<code>@mxdys</code>, shown to be a Cryptid by <code>@racheline</code>.
|Same as '''Hydra''' but starting iteration from 8 instead of 3 and with termination condition <code>O > 2E</code> instead of <code>E > 2O</code>, hence the name '''Antihydra'''.
|-
|[[Lucy's Moonlight]]
|[[BB(6)]]
|{{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}
|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|Aug 2024
|<code>mxdys</code>
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}
|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|Nov 2024
|Racheline
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}
|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|Jul 2024
|<code>mxdys</code>
|Has near-identical behavior to 16 related BB(6) holdouts. Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}}
|Jul 2024
|Racheline
|Probviously decidable. Estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.
|}
The following machines have chaotic behavior, but have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}} [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]

== Larger Cryptids ==

A more complete list of notable known Cryptids over a wider range of states and symbols. These Cryptids were all "constructed" rather than "discovered".

{| class="wikitable"
|-
! Name !! BB domain !! Machine !! Announcement !! Date !! Discoverer !! Note
|-
|[[Independence from ZFC|ZF]]
|BB(432)
|style="width:30%;word-break:break-word"|Wade's machine: https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e
|
|2025
|Wade, based on work by CatIsFluffy and O'Rear
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|-
|RH
|BB(744)
|style="width:30%;word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/riemann-matiyasevich-aaronson.nql
|
|2016
|Matiyasevich and O’Rear
|The machine halts if and only if [https://en.wikipedia.org/wiki/Riemann_hypothesis Riemann Hypothesis] is false.
|-
|Goldbach
|BB(25)
|style="width:30%;word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RD_1LC1RB_0RA1LC_0LQ1RE_0LF1RG_0LC1LF_0LF0LH_1LQ1LI_0RJ0LI_1RK0LJ_0RL0RS_1RL0RM_1RN1RM_0LO0LU_0LP1LO_1RH1LX_1LR1LQ_0RK0LT_1LR1RS_---1RC_1LV1LU_0LW0LJ_0RK0LW_1RY1LX_1RE1RY</pre></div></div>
|
|2016
|anonymous
|The machine halts if and only if [https://en.wikipedia.org/wiki/Goldbach%27s_conjecture Golbach's conjecture] is false. Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
| Erdős
| BB(5,4) and BB(15)
|style="width:30%;word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE</pre></div></div>
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><pre style="word-break:break-all">1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA</pre></div></div>
|| [https://arxiv.org/abs/2107.12475 arxiv preprint] || Jul 2021 || [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods || The machine halts if and only if the following conjecture by Erdős is false: "For all n > 8, there is at least one 2 in the base-3 representation of 2^n"
|-
|Weak Collatz
|BB(124) and BB(43,4)
|style="width:30%;word-break:break-word"|https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt (unverified)
https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt (unverified)
|
|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The machine halts if and only if the "weak Collatz conjecture" is false. The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers.
Not independently verified, and probably easy to further optimise.
|-
| Bigfoot - compiled|| [[BB(7)]]||style="width:30%;word-break:break-word"| <code>0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB</code>|| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment] || June 2024 || <code>@Iijil1</code>|| Compilation of Bigfoot into 2 symbols, there was a previous compilation [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 with 8 states]
|-
| Hydra - compiled
|BB(9)
|style="width:30%;word-break:break-word"|<pre>
0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ
</pre>[[File:Hydra_9_states.txt]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1251572501578780782 Discord message]
|June 2024
|<code>@Iijil1</code>
|Compilation of Hydra into 2 symbols, all [https://discord.com/channels/960643023006490684/1084047886494470185/1253193750486974464 confirmed by Shawn Ligocki]. <code>@Iijil1</code> provided 24 TMs which all emulate the same behavior.
<small>[https://discord.com/channels/960643023006490684/1084047886494470185/1247560072427474955 Previous compilation had 10 states], by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|}

== Beeping Busy Beaver ==

Cryptids were actually noticed in the [[Beeping Busy Beaver]] problem before they were in the classic Busy Beaver. See [[Mother of Giants]] describing a "family" of Turing machines which "[[probviously]]" [[quasihalt]], but requires solving a Collatz-like problem in order to actually prove it. They are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

Talk:1RB1LE 1LC0RA 0RF0LD 1LE1LA 1RC0LB ---1RC

2025-08-31T03:50:53Z

C7X: /* More simulation */

Turing machine for what function?
[[User:Qwerpiw|Qwerpiw]] ([[User talk:Qwerpiw|talk]]) 01:51, 7 August 2025 (UTC)

== More simulation ==

It looks like it writes at least 10^53 ones according to Quick_Sim.py, I can try more experiments by messing with the block finder [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 11:16, 30 August 2025 (UTC)

Edit: no halt yet at >10^261 ones, >10^527 steps [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 03:50, 31 August 2025 (UTC)

Talk:1RB1LE 1LC0RA 0RF0LD 1LE1LA 1RC0LB ---1RC

2025-08-30T11:16:49Z

C7X: /* More simulation */

Talk:1RB1LE 1LC0RA 0RF0LD 1LE1LA 1RC0LB ---1RC

2025-08-30T11:16:02Z

C7X: /* More simulation */ new section

Turing machine for what function?
[[User:Qwerpiw|Qwerpiw]] ([[User talk:Qwerpiw|talk]]) 01:51, 7 August 2025 (UTC)

== More simulation ==

It looks like it writes at least 10^53 nonzeros according to Quick_Sim.py, I can try more experiments by messing with the block finder [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 11:16, 30 August 2025 (UTC)

Maximum Consecutive Ones Function

2025-08-15T01:06:04Z

C7X: Add an early appearance of the num(5) champion /* Champions */

The '''Maximum Consecutive Ones''' function (named <math>num(n)</math> by Ben-Amram, Julstrom and Zwick)<ref>Ben-Amram A.M., Julstrom B.A. and Zwick U. (1996). [http://dx.doi.org/10.1007/BF01192693 A note on busy beavers and other creatures]. ''Mathematical Systems Theory'' '''29''' (4), July-August 1996, 375-386.</ref> is a [[Busy Beaver function]] which measures the maximum number of consecutive 1s left on the tape at halt across all n-state 2-symbol [[Turing machine]]s which leave all their 1s consecutively. Unlike <math>\Sigma(n)</math>, this allows some amount of order over the "API" of these TMs, so that their output can be used as inputs to another TM in some deterministic fashion. Note however, that this definition does not stipulate where the TM head must be positioned relative to the sequence of consecutive ones, so it is not completely trivial to use these results as input to a second TM.

This function is only defined for 2-symbol TMs. It is not entirely clear how best to extend it to multi-symbol TMs. One option would be to keep the definition the same, so only TMs which halt with all 1s on the tape would qualify. Another would be to allow any symbols as long as all non-0 symbols were in one contiguous chunk on the tape.

== Champions ==

{| class="wikitable"
|+ Known Values of num(n) function
|-
! Domain !! <math>num(n)</math> !! Champion TM !! Halting Config
!Source
|-
| [[BB(1)]]|| 1 || {{TM|1RZ---}} || <math>0^\infty \; 1 \; \textrm{Z>} \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(2)]]
|4
|{{TM|1RB1LB_1LA1LZ}}
|<math>0^\infty \; 1 \; \textrm{<Z} \; 1^3 \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(3)]]
|6
|{{TM|1RB1LC_1RC1LZ_1LA0LB}} and 3 others
|<math>0^\infty \; 1 \; \textrm{<Z} \; 1^5 \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(4)]]
|12
|{{TM|1RB0LA_1RC1LB_1LB1RD_1RZ0RA}}
|<math>0^\infty \; 1^{12} \; \textrm{Z>} \; 0^\infty</math>
|Ben-Amram, Julstrom and Zwick (1996)
|-
|[[BB(5)]]
|165
|{{TM|1RB1LA_1RC1LE_1RD1RE_0LA1RC_1RZ0LB}}
|<math>0^\infty \; 1^{165} \; \textrm{Z>} \; 0^\infty</math>
|[https://groups.google.com/g/busy-beaver-discuss/c/pvtXPmuMsAg/m/UP0xcmwoEAAJ Announcement] by Andrés Sancho in Feb 2025
|}
Note: All TMs here are listed in [[TNF-1RB]] format with the exception that sometimes the halt transition is chosen to be <code>1LZ</code> when that leads to a "simpler" halting configuration.

For BB(3), there are actually 4 champion machines that all leave exactly 6 1s: {{TM|1RB1RZ_0RC1RB_1LC1LA}}, {{TM|1RB1LC_1LA1RB_1LB1RZ}}, {{TM|1RB1RA_1LC1RZ_1RA1LB}}, {{TM|1RB1LC_1RC1RZ_1LA0LB}}.

The num(5) champion was found in 2009 by Joachim Hertel.<ref>J. Hertel, "[https://content.wolfram.com/sites/19/2009/11/Hertel.pdf Computing the Uncomputable Rado Sigma Function]". The Mathematica Journal vol. 11 (2009).</ref><sup>p. 282</sup>

== References ==
<references />
[[category:Functions]]