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2026-11-01T09:28:15Z

Frogmatic: Clarify the table key

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

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

The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The Busy Beaver function is not [[wikipedia:Computability|computable]], but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
| style="background: orange;" title="Cryptids: Antihydra and some other cryptids" | [[BB(6)]] > <math>10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
| style="background: #ddc290;" | [[BB(7)]] > <math>3 \uparrow^{20} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" title="Cryptid: Bigfoot" | [[BB(3,3)]] <math>\ge</math> 119,112,334,170,342,541
| style="background: #ddc290;" | [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math>
| style="background: #ddc290;" |
| style="background: #ddc290;" | BB(6,3) > <math> f_\omega^2(10\uparrow\uparrow\uparrow 6)</math>
| style="background: #ddc290;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ddc290;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
| style="background: #ddc290;" | [[BB(4,4)]] > <math>2 \uparrow^{509} 3</math>
| style="background: #ddc290;" | BB(5,4) > <math>f_\omega^2(10\uparrow^4 10\uparrow\uparrow 65\,533)</math>
| style="background: yellow;" | BB(6,4) > <math> f_{\omega+1}(10\uparrow\uparrow 65\,533)</math>
| style="background: yellow;" |
|-
! 5-symbol
| style="background: orange;" title="Cryptids: Hydra and Bonus Cryptid" | [[BB(2,5)]] > <math>10^{10^{10^{3\,314\,360}}}</math>
| style="background: #ddc290;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: yellow;" | BB(4,5) > <math> f_{\omega+1}(27\,041)</math>
| style="background: yellow;" |BB(5,5) > <math>f_{\omega+1}(f_\omega^4(6))</math>
| style="background: yellow;" | BB(6,5) > <math>f_{\omega+3}(11)</math>
| style="background: yellow;" |BB(7,5) > <math>f_{\omega+3}^2(10)</math>
|-
! 6-symbol
| style="background: #ddc290;" | [[BB(2,6)]] > <math>10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}}</math>
| style="background: #ddc290;" | BB(3,6) > <math> f_\omega^3(15)</math>
| style="background: yellow;" | BB(4,6) > <math> f_{\omega+1}(2 \uparrow^{15} 5)</math>
| style="background: yellow;" |
| style="background: yellow;" |
| style="background: yellow;" |
|}

'''Table key:'''

* <span style="background: orange">Orange cell</span>: smallest known [[Cryptid]] (a mathematically-hard machine) is in this class
* <span style="background: #ddc290">Tan cell</span>: a Cryptid is known using fewer states or symbols than this class.
* <span style="background: yellow">Yellow cell</span>: the Busy Beaver number is known to be more than [[Graham's number]].
* Blank cell: no known champion that surpasses a champion with fewer states or symbols by a non-trivial amount.

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

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

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

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

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

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

[[TMBR: May 2026]], [[TMBR: June 2026]], [[TMBR: July 2026]], [[TMBR: August 2026]], [[TMBR: September 2026]], and [[TMBR: October 2026]] are currently works in progress.

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

==Notes==
<references />

BBBB

2026-11-01T08:00:40Z

Frogmatic: Redirected page to Beeping Busy Beaver#Beeping Booping Busy Beavers

#redirect [[Beeping Busy Beaver#Beeping Booping Busy Beavers]]

Champions

2026-10-31T18:33:21Z

Frogmatic: /* 2-Symbol TMs */ Tidy up absent values

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]]; see [[measuring large numbers]] for more information. Machines below 6 states are listed in [[Tree Normal Form|TNF]] format, where machines which tie for champion are all listed. Note that most champions above 6 states are self-reported and have not been independently verified.



{| class="wikitable"
|+
!Domain
!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>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|mxdys in 2025
|See mxdys's analysis on the TM page
|-
|[[BB(7)]]
|<math>> 3 \uparrow^{20} 3 > f_\omega(20)</math>
|{{TM|1RB0RG_1RC1RB_0RD1RZ_0LE0LD_0LF1LE_1RA1LF_1LC0RB|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB7/1RB0RG_1RC1RB_0RD---_0LE0LD_0LF1LE_1RA1LF_0LD0RB.md Claimed analysis by Claude] never verified.
|-
|[[BB(8)]]
|<math>> f_\omega^3(12)</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG1RZ|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB8/1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG---.md Claimed analysis by Claude] never verified.
|-
|BB(9)
|<math>> f_\omega^7(12)</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI1RZ_1RI1RC|halt}}
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB9/1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI---_1RI1RC.md Claimed analysis by Claude] never verified.
|-
|BB(10)
|<math>> f_{\omega+1}(2 \uparrow^{36} 3) > g_{64}</math>
|<span style="word-break:break-all">{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RZ0LC|halt}}</span>
|[https://discord.com/channels/960643023006490684/1331570843829932063/1557806386073702622 Jeff Ketchersid and Claude in 10/2026 (clamied)]
|[https://github.com/jeff87654/busy-beaver/tree/main/BB10 Claimed analysis by Claude] never verified.
|-
|BB(11)
|<math>> f_{\omega+1}(f_\omega^2(5))</math>
|<span style="word-break:break-all">{{TM|1RG0LH_1RK1RC_0LD1LA_0RE1LD_0LI1RF_1RZ0RG_1RB1RG_1LA1LH_0LK0LJ_1LH1LI_1LJ1RE|halt}}</span>
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB11 Claimed analysis by Claude] never verified.
|-
|BB(12)
|<math>> f_{\omega+2}(f_{\omega+1}^2(3))</math>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK|halt}}</span>
|Jeff Ketchersid and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB12 Claimed analysis by Claude] never verified.
|-
|BB(13)
|<math>> f_{\omega+2}^6(f_{\omega+1}^2(3))</math>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LL0LL_0LH1LK_1RZ0LH|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB13 Claimed analysis by Claude] never verified.
|-
|BB(14)
|<math>> f_{\omega+2}^9(f_{\omega+1}^2(3))</math>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RM0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_0LN0LL_0LH1LK_1RZ0LH_0RJ0LI|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB14 Claimed analysis by Claude] never verified.
|-
|colspan="5"|...
|-
|BB(16)
|<math>> f_{\omega+2}^{10}(9)</math>
|<span style="word-break:break-all">{{TM|1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_1LG1RI_1RM0LM_1LG0RC_0RN0LE_1LA1LE_0LL1LB_0RM1LL_1LF1RH_0RO0LO_1LP1RM_1RZ1LI|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB16 Claimed analysis by Claude] never verified.
|-
|colspan="5"|...
|-
|BB(18)
|<math>> f_{\omega+3}(114)</math>
|<span style="word-break:break-all">{{TM|1LB1LA_1RC0LA_1RD1RC_1RE1RK_1RF0LJ_1LG1RI_1RM0LM_1LG0RC_1RN0LE_1LA1LE_0LL1LB_0RM1LL_1LF1RH_0RQ0RG_0LG1LP_0LO1LP_1LQ0RR_1RZ0LP|halt}}</span>
|Jeff Ketchersid, Claude and Gemini in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB16 Claimed analysis by Claude] never verified.
|-
|BB(19)
|
|
|
|
|-
|BB(20)
|<math>> f_{\omega + 6}^2(4)</math>
|<span style="word-break:break-all">{{TM|1RR---_0LJ0RC_1RB1LD_0LD1LE_1LF1RG_1LI0LE_0RE1LD_1RB1RH_1RH1RN_1LK1LJ_0LL1LJ_1RM1RA_0RM0RH_0LO1RN_1RH0LP_1RH1LQ_1RH1LO_1LS1RR_1RZ1LT_0LH0LS|halt}}</span>
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|<span style="word-break:break-all">{{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LR|halt}}</span>
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|Not independently verified.
|-
|colspan="5"|...
|-
|BB(29)
|<math>> f_{\omega^\omega}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB29/hydra29.md]
|mxdys in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB29 Claimed analysis by Claude] never verified.
|-
|BB(31)
|<math>> f_{\omega \uparrow \uparrow 4}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB31/hydra31.md]
|mxdys in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB31 Claimed analysis by Claude] never verified.
|-
|BB(33)
|<math>> f_{\omega \uparrow \uparrow 6}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB33/hydra33.md]
|mxdys in 10/2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB33 Claimed analysis by Claude] never verified.
|-
|BB(51)
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|[[Talk:Champions|See talk page]]
|[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Racheline in 2024]
|Not independently verified.
|-
|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>
|Not independently verified.
|-
|BB(1015)
|<math>> D^{5}(99)</math> = Loader's Number
|
|CatsAreFluffy in 2024<ref>https://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe</ref>
|Not independently verified.
|}

=== Conjectured ===

{| class="wikitable"
|+
!Domain
!Runtime
!Champions
!Discovered by
!Notes
|-
|BB(67)
|<math>> q^{81}(q(5)-2)</math>
|[https://wiki.bbchallenge.org/wiki/Talk:Champions]
|[https://discord.com/channels/960643023006490684/1331570843829932063/1339669138510970881 racheline in 2025]
|<math>q(n)</math> is derived from [https://en.wikipedia.org/wiki/Laver%20table Laver tables]. From [https://discord.com/channels/960643023006490684/960643023530762341/1525489299032641657 racheline]: "The known information about q(n) isn't enough to prove that this machine scores higher than the shown champions with fewer states, or that it scores lower than the shown champions with more states."
|}

== 3-Symbol TMs ==

{| class="wikitable"
|+
!Domain
!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
|See Polygon's analysis on the TM page
|-
|BB(5,3)
|
|
|
|
|-
|BB(6,3)
|<math>> f_\omega^2(10\uparrow\uparrow\uparrow 6)</math>
|{{TM|1LE2LD2LC_2RB1RB1RA_2RB1LC0LC_1RB1LD2LD_2RD2LE2LF_1RZ---1LE|halt}}
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(7,3)
|
|
|
|
|-
|BB(8,3)
|<math>> f_{\omega+1}(f_\omega^4(f_7(2))) > g_{64}</math>
|<span style="word-break:break-all">{{TM|1RD2RE2LA_2RC2LC2RB_1LH2RF1LA_2RC1RD2RB_---1LC2RE_1RZ0LF1LG_1RD2LG2LG_0LA0LD1LH|halt}}</span>
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(9,3)
|
|
|
|
|-
|BB(10,3)
|
|
|
|
|-
|BB(11,3)
|<math>> f_{\omega+3}^2(4)</math>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_2LE1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI---_0RC1LH2RG_1RC2LH1LA_1RI1RI1LJ_0LA2LK---_1RZ2LJ---|halt}}</span>
|Jacobzheng in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(36,3)
|<math>> f_{\varepsilon_0}(7)</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(36,3)</ref>
|Not independently verified.
|-
|BB(37,3)
|<math>> f_{\varepsilon_0}(373\,676\,378)</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(37,3)</ref>
|Not independently verified.
|}

== 4-Symbol TMs ==
{| class="wikitable"
|+
!Domain
!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]
|-
|[[BB(4,4)]]
|<math>> 2 \uparrow^{509} 3 > f_\omega(508)</math>
|{{TM|0LB3LB1RZ2RA_1LD3RB1LC2RA_3RB1LB3LC2RC_0RD2RC2LB0RA|halt}}
|[[User:Loader3229]] in 2026
|[https://github.com/loader3229/busy-beaver/tree/main/BB44 Claimed analysis by MiniMax] Not independently verified.
|-
|BB(5,4)
|<math>> f_\omega^2(10\uparrow^4 10\uparrow\uparrow 65\,533)</math>
|<span style="word-break:break-all">{{TM|1LD2RA3RB1LC_1LD2LA3RB---_2RA1LA3RC2LC_2RA---2LD2LE_------1RZ1LD|halt}}</span>
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(6,4)
|<math>> f_{\omega+1}(10\uparrow\uparrow 65\,533) > g_{64}</math>
|<span style="word-break:break-all">{{TM|1LF2RA3RB---_3RD2LB3RB1LC_2RA1LB3RC2LC_1LD2RE0LA1LD_1RZ0LE1LF---_2RA---2LF2LF|halt}}</span>
|Jacobzheng in 2026
|Not independently verified.
|-
|BB(7,4)
|
|
|
|
|-
|BB(8,4)
|<math>> f_{\omega+2}(f_\omega^4(12))</math>
|<span style="word-break:break-all">{{TM|1RC3LB2RD2RA_1RF3RB1LC2RA_3RB1LB3LC2RC_0RF2LD1LD2LE_3RE3LB2RE2LE_0LG0LH1RZ3RB_1LG2RD1LG3RG_0LH0LC0LH3RD|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(9,4)
|<math>> f_{\omega+3}(16)</math>
|<span style="word-break:break-all">{{TM|2LI3LB2RD2RA_2LC3RB1LC2RA_3RB1LB3LC2RC_0RF2LD1LD2LE_3RE3LB2RE2LE_0LG0LH1RZ---_1LG2RD1LG3RG_0LH---0LH3RD_2LB---1LI3RF|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(31,4)
|<math>> f_{\varepsilon_0}(373\,676\,378)</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(31,4)</ref>
|Not independently verified.
|}

== 5-Symbol TMs ==
{| class="wikitable"
!Domain
!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
|Not independently verified.
|-
|BB(4,5)
|<math>> f_{\omega+1}(27\,041) > g_{64}</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA4RD_3RB1LB3LC2RC4LC_2RC2RD1RZ4RB1LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(5,5)
|<math>> f_{\omega+1}(f_\omega^4(6))</math>
|{{TM|1LB3LB4LC2RA4LB_3LE3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ------4LD4LE_3RB------4LE3LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(6,5)
|<math>> f_{\omega+3}(11)</math>
|<span style="word-break:break-all">{{TM|2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LA4LD4LE_3RB---0RD4LE3LC_4LC------3LF3LF|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(7,5)
|<math>> f_{\omega+3}^2(10)</math>
|<span style="word-break:break-all">{{TM|2LE3LB4LC2RA4LB_4RA3RB1LC2RA4RD_3RB1LB3LC2RC4LC_1RZ3LF1LG4LD4LE_3RB---1LE4LE3LC_4LC------3LF3LF_3LF------1LG1LG|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(8,5)
|<math>> f_{\omega \times 2}(32)</math>
|<span style="word-break:break-all">{{TM|0LB3LB1RZ2RA4RB_0LD1RB1LD2RA1LE_3RB1LB3LC2RC3LF_3RE2LF3LC2RC4LF_0LF3RE0LG4LH2LB_0LE3LG4LC2RF4LG_1LC3RG1LH2RF2RA_3RG1LG3LH2RH4LH|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(19,5)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<span style="word-break:break-all">{{TM|1RB2LA1RA2LB2LA_0LA2RB3RB4RA1LQ_1RZ1RC2LD------_0LK3RD2RD2RD4RD_------0RH1LS0RF_1LG1RF2RF---4RF_4LE1LG2LG---4LG_0LI1RH2RH---4RH_0RK0LJ------4RK_2LE1LJ2LJ---4LJ_4RP---0LE---0LK_2RQ---2LL---4LL_2LN---2RM3RM4RM_4RQ2RQ2LN3LN4LN_4LN---2RO3RO4RO_3LL1LP2LP1LQ4LP_1RC3RR1RM3LP0RO_3LL1RR2RR0RR4RR_1RC1LS2LS1LS4LS|halt}}</span>
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(19,5)</ref>
|Not independently verified.
|-
|BB(94,5)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|Not independently verified.
|}

== 6-Symbol TMs ==
{| class="wikitable"
!Domain
!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]
|-
|BB(3,6)
|<math>> f_\omega^3(15)</math>
|{{TM|1RB3LB4LC2RA4LB5LB_2LC3RB1LC2RA5LC1RZ_3RB1LB3LC2RC4LC5LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(4,6)
|<math>> f_{\omega+1}(2 \uparrow^{15} 5)</math>
|{{TM|1RB3LB5LD2RA4LB5LB_2LC3RB1LC2RA3LC1RZ_3RB1LB3LC2RC4LC5LC_3RC---4LD4LD------|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(61,6)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|Not independently verified.
|}

== 7-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,7)]]
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
|{{TM|1RB3RB4LA1LB5LA6LB5RA_2LA6RA4RB1RZ3LB6LA4RB|halt}}
|[https://discord.com/channels/960643023006490684/960643023530762341/1563057558011252761 roromachine14 in 2026]
|Not independently verified.
|-
|BB(3,7)
|<math>> f_\omega^4(15)</math>
|{{TM|1RB3LB4LC2RA4LB5LB6LB_2LC3RB1LC2RA5LC6LC1RZ_3RB1LB3LC2RC4LC5LC6LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(18,7)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<span style="word-break:break-all">{{TM|1RB2LA1RA2LB2LA------_0LA2RB3RB4RA1LP------_0RZ1RC2LD------------_0LJ3RD2RD2RD4RD------_4LE1LE0RG1LR0RF2LE4LE_1LE1RF5RF---6RF------_0LH1RG2RG---4RG------_0RJ0LI------4RJ------_2LE1LI2LI---4LI------_4RO---0LE---0LJ------_2RP---2LK---4LK------_2LM---2RL3RL4RL------_4RP2RP2LM3LM4LM------_4LM---2RN3RN4RN------_3LK1LO2LO1LP4LO------_1RC3RQ1RL3LO0RN------_3LK1RQ2RQ0RQ4RQ------_1RC1LR2LR1LR4LR------|halt}}</span>
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(18,7)</ref>
|Not independently verified.
|-
|BB(46,7)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|Not independently verified.
|-
|BB(134,7)
|<math>> f_{\psi_0(\Omega_\omega)}(2\,050)</math>
|
|LittlePeng9+Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/Analysis_of_LittlePeng9%27s_work/Hydra_machines</ref>
|Not independently verified.
|}

== 8-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|BB(2,8)
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
|{{TM|1RB3RB4LA1LB5LA6LB5RA1RZ_2LA6RA4RB7RA3LB6LA4RB4RA|halt}}
|[https://discord.com/channels/960643023006490684/1084047886494470185/1563070360956698804 roromachine14 in 2026]
|Not independently verified.
|-
|BB(3,8)
|<math>> f_\omega^5(15)</math>
|{{TM|1RB3LB4LC2RA4LB5LB6LB7LB_2LC3RB1LC2RA5LC6LC7LC1RZ_3RB1LB3LC2RC4LC5LC6LC7LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(17,8)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<span style="word-break:break-all">{{TM|1RB2LA1RA2LB2LA---------_0LA2RB3RB4RA1LO---------_0RZ1RC2LD---------------_0LJ3RD2RD2RD4RD---------_4LE1LE0RG1LQ0RF2LE4LE---_1LE1RF5RF---6RF---------_0LH1RG2RG---4RG---------_0RJ0LI------4RJ---------_2LE1LI2LI---4LI---------_4RN---0LE---0LJ---------_2RO2RO2LK3LK4LK2LK4LK4RO_2LK---5RL3RL6RL---------_4LK---5RM3RM6RM---------_3LK1LN2LN1LO4LN---------_1RC3RP1RL3LN7RM---------_3LK1RP2RP0RP4RP---------_1RC1LQ2LQ1LQ4LQ---------|halt}}</span>
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(17,8)</ref>
|Not independently verified.
|}

== 9-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|BB(2,9)
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
|{{TM|1RB3RB4LA1LB5LA6LB5RA8RA1RZ_2LA6RA4RB7RA3LB6LA4RB4RA4RA|halt}}
|[https://discord.com/channels/960643023006490684/1084047886494470185/1563070360956698804 roromachine14 in 2026]
|Not independently verified.
|-
|BB(3,9)
|<math>> f_\omega^6(15)</math>
|<span style="word-break:break-all">{{TM|1RB3LB4LC2RA4LB5LB6LB7LB8LB_2LC3RB1LC2RA5LC6LC7LC8LC1RZ_3RB1LB3LC2RC4LC5LC6LC7LC8LC|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(42,9)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|Not independently verified.
|}

== Zoology ==
There are many types of champions.

Note: the scale here is not formally defined. It mostly serves to estimate the average values produced by a type of champions.
{| class="wikitable"
|+
!Classification
!Description
!Examples
!Scale
|-
|Trivial
|The simplest champions that can exist. They mostly appear in some BB-adjacent functions like [[Fractran|BBf]] or [[CounterScript|BBCS]].
|
* {{TM|1RZ---|halt}}
* {{TM|0RA_0RZ|halt}}
* {{TM|0RA_0RB_0RZ|halt}}
|<math>O(n)</math>
|-
|Chaotic
|Have a chaotic behavior with repeating patterns that go back and forth.
|
* {{TM|1RB1LB_1LA1RZ|halt}}
* {{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
* {{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|<math>O(2^n)</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|1RB2LB1RZ_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_1LB1LA3RB1RZ|halt}}
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|<math>O(2^n)</math>
|-
|Random walk
|Compute a [[Collatz-like]] function with a random walk.
|
|<math>O(2^{2^n})</math>
|-
|Tetrational
|
|
|<math>O(2 \uparrow\uparrow n)</math>
|-
|Ackermannian
|
|
|<math>O(2 \uparrow^n 3)</math>
|}

== References ==

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

Beeping Busy Beaver

2026-10-31T13:47:14Z

Frogmatic: /* Champions */ Add the best source I can find for BBB(3) = 55.

The '''Beeping Busy Beaver''' (BBB) function is a variant of the [[Busy Beaver Functions|Busy Beaver function]] proposed by Scott Aaronson in his 2020 [[Busy Beaver Frontier]] survey.<ref name=":0">Scott Aaronson. "The Busy Beaver Frontier". https://www.scottaaronson.com/papers/bb.pdf</ref> It is notable because it is uncomputable even if you have access to a halting oracle for Turing Machines. These can simulate problems in Π₂ and Σ₂ of the [[wikipedia:Arithmetic hierarchy|arithmetic hierarchy]], such as the [[wikipedia:Collatz conjecture|Collatz conjecture]] and [[wikipedia:Twin prime#Twin prime conjecture|twin prime conjecture]].

== Definition ==
Consider a ''beeping Turing machine'', which is a [[Turing machine]] that has a special state named "beep state". Every time the TM enters the "beep state" it ''beeps''. There are two possibilities, either this TM beeps a finite number of times (and thus there is a final beep) or it never stops beeping. Nick Drozd coined the term [[Quasihalt|quasihalting]] to describe the event when a TM last beeps. A TM quasihalts if it beeps only a finite number of times.<ref>Nick Drozd. 2020. [https://nickdrozd.github.io/2020/08/13/beeping-busy-beavers.html Beeping Busy Beavers].</ref> The Beeping Busy Beaver problem is analogous to the Busy Beaver problem, replacing halting with quasihalting. In other words, let <math>b(M)</math> be the number of steps the machine M takes until it quasihalts (beeps for the last time) if it quasihalts (we will say <math>b(M) = \infty</math> if the TM never stops beeping). Then

<math display="block">\operatorname{BBB}(n) := \max_{{M \in T(n)\ :\ b(M) < \infty}} b(M)</math>

where <math>T(n)</math> is the set of Turing machines with <math>n</math> states and two symbols.

Note that quasihalting does not require halting, so we can consider Turing machines which have no halt transitions. Thus the [[Tree Normal Form]] algorithm needs to be modified (to allow TMs with no halt transitions) when searching for BBB champions.

Another way to define BBB is to have Turing machines have a special beep transition instead of an entire beep state.

== Growth rate ==

It is easy to see that <math>\operatorname{BBB}(n) \ge \operatorname{BB}(n)</math>, by letting the beep state be the state that is reached immediately before the halt state.

In fact, BBB grows much faster than BB. BBB eventually dominates any computable function augmented with an oracle for computing BB. So, for example, there exists some N such that for all n > N:

* <math>BBB(n) > BB(BB(n))</math>
* <math>BBB(n) > BB^n(n)</math>, where <math>BB^k</math> represents k iterations of BB
* <math>BBB(n) > BB^{BB(n)}(n)</math>

etc.

== Champions ==
This table considers beep states.
{| class="wikitable"
|+
!Domain
!Max Steps
!Champions
!Notes
|-
|BBB(1)
|1
|{{TM|1RZ---|halt}}
|Case analysis by Scott Aaronson.<ref name=":0" />
|-
|BBB(2)
|6
|{{TM|1RB1LB_1LB1LA|non_halt}}
|Case analysis by Scott Aaronson.<ref name=":0" />
|-
|BBB(3)
|55
|{{TM|1RB0LB_1LA0RC_1LC1LA|non_halt}}
|Claimed as a lower bound by Scott Aaronson, verified with exhaustive search by Nick Drozd.<ref name=":0" /><ref>Nick Drozd blog comment, 2021. https://scottaaronson.blog/?p=5661#comment-1900343</ref>
|-
|BBB(4)
|32,779,478
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD|non_halt}}
|Proven in Coq.<ref name=":Coq-BBB4">[https://github.com/carrino/Coq-BBB4 Coq-BBB4]</ref>
|-
|BBB(5)
|> 10<sup>14,006</sup>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA|non_halt}}
|Best known, per Nick Drozd.<ref>Nick Drozd. https://scottaaronson.blog/?p=8088#comment-1981333</ref> Probably ><math>10^{10^{10^5}}</math> due to four [[probviously]] quasihalting [[Cryptids]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/04/03/mother-of-giants.html Mother of Giants].</ref>
|-
|BBB(2,3)
|59
|{{TM|1RB2LB1LA_2LB2RA0RA|non_halt}}
|Per Nick Drozd.<ref>Nick Drozd, "[https://nickdrozd.github.io/2025/03/24/bbb-3-3.html BBB(3,3) > 10↑↑6]". Blog post. Accessed 15 August 2025.</ref>
|-
|BBB(3,3)
|> 10 ↑↑ 6.74
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC|non_halt}}
|Best known, per Nick Drozd.<ref name=":Nick Drozd Tetrational Blank26">Nick Drozd, "[https://nickdrozd.github.io/2026/09/01/tetrational-self-cleaning-turing-machine.html A Tetrational Self-Cleaning Turing Machine]". Blog post. Accessed 17 October 2026.</ref>
|-
|BBB(2,4)
|> <math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA|non_halt}}
|Best known, per Nick Drozd.<ref>Nick Drozd. "[https://nickdrozd.github.io/2022/02/11/latest-beeping-busy-beaver-results.html Latest Beeping Busy Beaver Results]". Accessed 15 August 2025.</ref>
|}

All known champions quasihalt by becoming [[Translated Cycler|Translated Cyclers]], a property which is known to be weaker than the general quasihalting condition.

==Beeping Booping Busy Beavers==
A further extension devised by Bram Cohen is as follows: a Turing machine has two special transitions, a beep transition and a boop transition, both of which may repeat infinitely often. The machine "outputs" an integer sequence, indicated by the number of beeps between successive boops. If any such integer is repeated an infinite number of times, the machine is said to '''pseudoquasihalt'''. Now if we define <math>g(M)</math> as the number of transitions for machine M to finish outputting that infinitely-repeated value for the first time (or ∞ if there is no such value), the Beeping Booping Busy Beaver number is:

<math display="block">\operatorname{BBBB}(n) := \max_{{M \in T(n)\ :\ g(M) < \infty}} g(M)</math>

These machines are equivalent to Turing machines with second-order oracles, and so their pseudoquasihalting behavior can describe sets in Π₃ and Σ₃ of the [[wikipedia:Arithmetic hierarchy|arithmetic hierarchy]].<ref>Bram Cohen. 2023. [https://bramcohen.com/p/beeping-booping-busy-beavers Beeping Booping Busy Beavers].</ref>

There are multiple models to choose from, depending on whether one allows multiple beep transitions and multiple boop transitions, or only one of each, or only one boop transition while making every transition a beep transition. All of these models have the same computational strength and corresponding growth rate, but each has its own advantages. The first is the most general, including all machines from the other two, and in that sense it is more natural, but that makes the space of candidate machines to search through significantly larger. The second is the original definition and it is relatively simple. The third is equivalent to simply considering the amount of time between successive boops, instead of the number of beeps between successive boops, and has the smallest space of candidate machines. It is not yet decided which of these models (or perhaps a model different from all three) should be chosen for "the" BBBB function.

=== BBBB champions ===

{| class="wikitable"
|+
!Domain
!Value
!Champions
!Notes
|-
|BBBB(1)
|2
|{{TM|0RA---|non_halt}} (beep: any, boop: A0)
|Confirmed.<ref>racheline, September 2025. [https://discord.com/channels/960643023006490684/1243312334907375676/1417839422191308871 Discord message]</ref>
|-
|BBBB(2)
|18
|{{TM|0RB1LB_1LA1RB|non_halt}} (beep: A0, boop: A1)
|Confirmed by exhaustive search.<ref>Charlie Reams. October 2026. [https://discord.com/channels/960643023006490684/1243312334907375676/1565879387927224474 Discord message]</ref>
|-
|BBBB(3)
|≥ 127
|{{TM|1RB1LB_0RC0LA_1LC0LA|non_halt}} (beep: A0, boop: B1)
|Holdouts: 935.<ref name="calr3-bbbb">Charlie Reams, GitHub https://github.com/calr3/bbbb</ref>
|-
|BBBB(2, 3)
|≥ 187
|{{TM|2RB2LA1RA_1LB1LA0RB|non_halt}} (beep: B0, boop: B2)
|Holdouts: >2000.<ref name="calr3-bbbb"></ref>
|}

==Higher-order Beavers==
For higher order functions, using https://katelyndoucette.com/articles/building-the-busy-beaver-ladder definition, we can get BBBBB(1) = 2 and BBBBB(2) ≥ 18.

== References ==

[[category:Functions]]

Beeping Busy Beaver

2026-10-31T13:39:52Z

Frogmatic: /* Champions */ Split source to a separate column (it's hard to read a number with a footnote right after it, these look like exponents)

The '''Beeping Busy Beaver''' (BBB) function is a variant of the [[Busy Beaver Functions|Busy Beaver function]] proposed by Scott Aaronson in his 2020 [[Busy Beaver Frontier]] survey.<ref name=":0">Scott Aaronson. "The Busy Beaver Frontier". https://www.scottaaronson.com/papers/bb.pdf</ref> It is notable because it is uncomputable even if you have access to a halting oracle for Turing Machines. These can simulate problems in Π₂ and Σ₂ of the [[wikipedia:Arithmetic hierarchy|arithmetic hierarchy]], such as the [[wikipedia:Collatz conjecture|Collatz conjecture]] and [[wikipedia:Twin prime#Twin prime conjecture|twin prime conjecture]].

== Definition ==
Consider a ''beeping Turing machine'', which is a [[Turing machine]] that has a special state named "beep state". Every time the TM enters the "beep state" it ''beeps''. There are two possibilities, either this TM beeps a finite number of times (and thus there is a final beep) or it never stops beeping. Nick Drozd coined the term [[Quasihalt|quasihalting]] to describe the event when a TM last beeps. A TM quasihalts if it beeps only a finite number of times.<ref>Nick Drozd. 2020. [https://nickdrozd.github.io/2020/08/13/beeping-busy-beavers.html Beeping Busy Beavers].</ref> The Beeping Busy Beaver problem is analogous to the Busy Beaver problem, replacing halting with quasihalting. In other words, let <math>b(M)</math> be the number of steps the machine M takes until it quasihalts (beeps for the last time) if it quasihalts (we will say <math>b(M) = \infty</math> if the TM never stops beeping). Then

<math display="block">\operatorname{BBB}(n) := \max_{{M \in T(n)\ :\ b(M) < \infty}} b(M)</math>

where <math>T(n)</math> is the set of Turing machines with <math>n</math> states and two symbols.

Note that quasihalting does not require halting, so we can consider Turing machines which have no halt transitions. Thus the [[Tree Normal Form]] algorithm needs to be modified (to allow TMs with no halt transitions) when searching for BBB champions.

Another way to define BBB is to have Turing machines have a special beep transition instead of an entire beep state.

== Growth rate ==

It is easy to see that <math>\operatorname{BBB}(n) \ge \operatorname{BB}(n)</math>, by letting the beep state be the state that is reached immediately before the halt state.

In fact, BBB grows much faster than BB. BBB eventually dominates any computable function augmented with an oracle for computing BB. So, for example, there exists some N such that for all n > N:

* <math>BBB(n) > BB(BB(n))</math>
* <math>BBB(n) > BB^n(n)</math>, where <math>BB^k</math> represents k iterations of BB
* <math>BBB(n) > BB^{BB(n)}(n)</math>

etc.

== Champions ==
This table considers beep states.
{| class="wikitable"
|+
!Domain
!Max Steps
!Champions
!Notes
|-
|BBB(1)
|1
|{{TM|1RZ---|halt}}
|Case analysis by Scott Aaronson.<ref name=":0" />
|-
|BBB(2)
|6
|{{TM|1RB1LB_1LB1LA|non_halt}}
|Case analysis by Scott Aaronson.<ref name=":0" />
|-
|BBB(3)
|55
|{{TM|1RB0LB_1LA0RC_1LC1LA|non_halt}}
|Claimed as a lower bound by Scott Aaronson.<ref name=":0" />
|-
|BBB(4)
|32,779,478
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD|non_halt}}
|Proven in Coq.<ref name=":Coq-BBB4">[https://github.com/carrino/Coq-BBB4 Coq-BBB4]</ref>
|-
|BBB(5)
|> 10<sup>14,006</sup>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA|non_halt}}
|Best known, per Nick Drozd.<ref>Nick Drozd. https://scottaaronson.blog/?p=8088#comment-1981333</ref> Probably ><math>10^{10^{10^5}}</math> due to four [[probviously]] quasihalting [[Cryptids]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/04/03/mother-of-giants.html Mother of Giants].</ref>
|-
|BBB(2,3)
|59
|{{TM|1RB2LB1LA_2LB2RA0RA|non_halt}}
|Per Nick Drozd.<ref>Nick Drozd, "[https://nickdrozd.github.io/2025/03/24/bbb-3-3.html BBB(3,3) > 10↑↑6]". Blog post. Accessed 15 August 2025.</ref>
|-
|BBB(3,3)
|> 10 ↑↑ 6.74
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC|non_halt}}
|Best known, per Nick Drozd.<ref name=":Nick Drozd Tetrational Blank26">Nick Drozd, "[https://nickdrozd.github.io/2026/09/01/tetrational-self-cleaning-turing-machine.html A Tetrational Self-Cleaning Turing Machine]". Blog post. Accessed 17 October 2026.</ref>
|-
|BBB(2,4)
|> <math>2 \times 10^{23}</math>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA|non_halt}}
|Best known, per Nick Drozd.<ref>Nick Drozd. "[https://nickdrozd.github.io/2022/02/11/latest-beeping-busy-beaver-results.html Latest Beeping Busy Beaver Results]". Accessed 15 August 2025.</ref>
|}

All known champions quasihalt by becoming [[Translated Cycler|Translated Cyclers]], a property which is known to be weaker than the general quasihalting condition.

==Beeping Booping Busy Beavers==
A further extension devised by Bram Cohen is as follows: a Turing machine has two special transitions, a beep transition and a boop transition, both of which may repeat infinitely often. The machine "outputs" an integer sequence, indicated by the number of beeps between successive boops. If any such integer is repeated an infinite number of times, the machine is said to '''pseudoquasihalt'''. Now if we define <math>g(M)</math> as the number of transitions for machine M to finish outputting that infinitely-repeated value for the first time (or ∞ if there is no such value), the Beeping Booping Busy Beaver number is:

<math display="block">\operatorname{BBBB}(n) := \max_{{M \in T(n)\ :\ g(M) < \infty}} g(M)</math>

These machines are equivalent to Turing machines with second-order oracles, and so their pseudoquasihalting behavior can describe sets in Π₃ and Σ₃ of the [[wikipedia:Arithmetic hierarchy|arithmetic hierarchy]].<ref>Bram Cohen. 2023. [https://bramcohen.com/p/beeping-booping-busy-beavers Beeping Booping Busy Beavers].</ref>

There are multiple models to choose from, depending on whether one allows multiple beep transitions and multiple boop transitions, or only one of each, or only one boop transition while making every transition a beep transition. All of these models have the same computational strength and corresponding growth rate, but each has its own advantages. The first is the most general, including all machines from the other two, and in that sense it is more natural, but that makes the space of candidate machines to search through significantly larger. The second is the original definition and it is relatively simple. The third is equivalent to simply considering the amount of time between successive boops, instead of the number of beeps between successive boops, and has the smallest space of candidate machines. It is not yet decided which of these models (or perhaps a model different from all three) should be chosen for "the" BBBB function.

=== BBBB champions ===

{| class="wikitable"
|+
!Domain
!Value
!Champions
!Notes
|-
|BBBB(1)
|2
|{{TM|0RA---|non_halt}} (beep: any, boop: A0)
|Confirmed.<ref>racheline, September 2025. [https://discord.com/channels/960643023006490684/1243312334907375676/1417839422191308871 Discord message]</ref>
|-
|BBBB(2)
|18
|{{TM|0RB1LB_1LA1RB|non_halt}} (beep: A0, boop: A1)
|Confirmed by exhaustive search.<ref>Charlie Reams. October 2026. [https://discord.com/channels/960643023006490684/1243312334907375676/1565879387927224474 Discord message]</ref>
|-
|BBBB(3)
|≥ 127
|{{TM|1RB1LB_0RC0LA_1LC0LA|non_halt}} (beep: A0, boop: B1)
|Holdouts: 935.<ref name="calr3-bbbb">Charlie Reams, GitHub https://github.com/calr3/bbbb</ref>
|-
|BBBB(2, 3)
|≥ 187
|{{TM|2RB2LA1RA_1LB1LA0RB|non_halt}} (beep: B0, boop: B2)
|Holdouts: >2000.<ref name="calr3-bbbb"></ref>
|}

==Higher-order Beavers==
For higher order functions, using https://katelyndoucette.com/articles/building-the-busy-beaver-ladder definition, we can get BBBBB(1) = 2 and BBBBB(2) ≥ 18.

== References ==

[[category:Functions]]

Beeping Busy Beaver

2026-10-31T13:11:45Z

Frogmatic: /* Beeping Booping Busy Beavers */ Document current BBBB champs for small values, and give proper credit to racheline.

The '''Beeping Busy Beaver''' (BBB) function is a variant of the [[Busy Beaver Functions|Busy Beaver function]] proposed by Scott Aaronson in his 2020 [[Busy Beaver Frontier]] survey.<ref name=":0">Scott Aaronson. "The Busy Beaver Frontier". https://www.scottaaronson.com/papers/bb.pdf</ref> It is notable because it is uncomputable even if you have access to a halting oracle for Turing Machines. These can simulate problems in Π₂ and Σ₂ of the [[wikipedia:Arithmetic hierarchy|arithmetic hierarchy]], such as the [[wikipedia:Collatz conjecture|Collatz conjecture]] and [[wikipedia:Twin prime#Twin prime conjecture|twin prime conjecture]].

== Definition ==
Consider a ''beeping Turing machine'', which is a [[Turing machine]] that has a special state named "beep state". Every time the TM enters the "beep state" it ''beeps''. There are two possibilities, either this TM beeps a finite number of times (and thus there is a final beep) or it never stops beeping. Nick Drozd coined the term [[Quasihalt|quasihalting]] to describe the event when a TM last beeps. A TM quasihalts if it beeps only a finite number of times.<ref>Nick Drozd. 2020. [https://nickdrozd.github.io/2020/08/13/beeping-busy-beavers.html Beeping Busy Beavers].</ref> The Beeping Busy Beaver problem is analogous to the Busy Beaver problem, replacing halting with quasihalting. In other words, let <math>b(M)</math> be the number of steps the machine M takes until it quasihalts (beeps for the last time) if it quasihalts (we will say <math>b(M) = \infty</math> if the TM never stops beeping). Then

<math display="block">\operatorname{BBB}(n) := \max_{{M \in T(n)\ :\ b(M) < \infty}} b(M)</math>

where <math>T(n)</math> is the set of Turing machines with <math>n</math> states and two symbols.

Note that quasihalting does not require halting, so we can consider Turing machines which have no halt transitions. Thus the [[Tree Normal Form]] algorithm needs to be modified (to allow TMs with no halt transitions) when searching for BBB champions.

Another way to define BBB is to have Turing machines have a special beep transition instead of an entire beep state.

== Growth rate ==

It is easy to see that <math>\operatorname{BBB}(n) \ge \operatorname{BB}(n)</math>, by letting the beep state be the state that is reached immediately before the halt state.

In fact, BBB grows much faster than BB. BBB eventually dominates any computable function augmented with an oracle for computing BB. So, for example, there exists some N such that for all n > N:

* <math>BBB(n) > BB(BB(n))</math>
* <math>BBB(n) > BB^n(n)</math>, where <math>BB^k</math> represents k iterations of BB
* <math>BBB(n) > BB^{BB(n)}(n)</math>

etc.

== Champions ==
This table considers beep states.
{| class="wikitable"
|+
!Domain
!Max Steps
!Champions
|-
|BBB(1)
|= 1<ref name=":0" />
|{{TM|1RZ---|halt}}
|-
|BBB(2)
|= 6<ref name=":0" />
|{{TM|1RB1LB_1LB1LA|non_halt}}
|-
|BBB(3)
|= 55
|{{TM|1RB0LB_1LA0RC_1LC1LA|non_halt}}
|-
|BBB(4)
|= 32,779,478<ref name=":Coq-BBB4">[https://github.com/carrino/Coq-BBB4 Coq-BBB4]</ref>
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD|non_halt}}
|-
|BBB(5)*
|> 10<sup>14,006</sup><ref>Nick Drozd. https://scottaaronson.blog/?p=8088#comment-1981333</ref>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA|non_halt}}
|-
|BBB(2,3)
|59<ref>Nick Drozd, "[https://nickdrozd.github.io/2025/03/24/bbb-3-3.html BBB(3,3) > 10↑↑6]". Blog post. Accessed 15 August 2025.</ref>
|{{TM|1RB2LB1LA_2LB2RA0RA|non_halt}}
|-
|BBB(3,3)
|> 10 ↑↑ 6.74<ref name=":Nick Drozd Tetrational Blank26">Nick Drozd, "[https://nickdrozd.github.io/2026/09/01/tetrational-self-cleaning-turing-machine.html A Tetrational Self-Cleaning Turing Machine]". Blog post. Accessed 17 October 2026.</ref>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC|non_halt}}
|-
|BBB(2,4)
|> <math>2 \times 10^{23}</math><ref>Nick Drozd. "[https://nickdrozd.github.io/2022/02/11/latest-beeping-busy-beaver-results.html Latest Beeping Busy Beaver Results]". Accessed 15 August 2025.</ref>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA|non_halt}}
|}
<nowiki>*</nowiki> probably <math>BBB(5) > 10^{10^{10^5}}</math> due to four "[[probviously]]" quasihalting [[Cryptids]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/04/03/mother-of-giants.html Mother of Giants].</ref>

All known champions quasihalt by becoming [[Translated Cycler|Translated Cyclers]], a property which is known to be weaker than the general quasihalting condition.

==Beeping Booping Busy Beavers==
A further extension devised by Bram Cohen is as follows: a Turing machine has two special transitions, a beep transition and a boop transition, both of which may repeat infinitely often. The machine "outputs" an integer sequence, indicated by the number of beeps between successive boops. If any such integer is repeated an infinite number of times, the machine is said to '''pseudoquasihalt'''. Now if we define <math>g(M)</math> as the number of transitions for machine M to finish outputting that infinitely-repeated value for the first time (or ∞ if there is no such value), the Beeping Booping Busy Beaver number is:

<math display="block">\operatorname{BBBB}(n) := \max_{{M \in T(n)\ :\ g(M) < \infty}} g(M)</math>

These machines are equivalent to Turing machines with second-order oracles, and so their pseudoquasihalting behavior can describe sets in Π₃ and Σ₃ of the [[wikipedia:Arithmetic hierarchy|arithmetic hierarchy]].<ref>Bram Cohen. 2023. [https://bramcohen.com/p/beeping-booping-busy-beavers Beeping Booping Busy Beavers].</ref>

There are multiple models to choose from, depending on whether one allows multiple beep transitions and multiple boop transitions, or only one of each, or only one boop transition while making every transition a beep transition. All of these models have the same computational strength and corresponding growth rate, but each has its own advantages. The first is the most general, including all machines from the other two, and in that sense it is more natural, but that makes the space of candidate machines to search through significantly larger. The second is the original definition and it is relatively simple. The third is equivalent to simply considering the amount of time between successive boops, instead of the number of beeps between successive boops, and has the smallest space of candidate machines. It is not yet decided which of these models (or perhaps a model different from all three) should be chosen for "the" BBBB function.

=== BBBB champions ===

{| class="wikitable"
|+
!Domain
!Value
!Champions
!Notes
|-
|BBBB(1)
|2
|{{TM|0RA---|non_halt}} (beep: any, boop: A0)
|Confirmed.<ref>racheline, September 2025. [https://discord.com/channels/960643023006490684/1243312334907375676/1417839422191308871 Discord message]</ref>
|-
|BBBB(2)
|18
|{{TM|0RB1LB_1LA1RB|non_halt}} (beep: A0, boop: A1)
|Confirmed by exhaustive search.<ref>Charlie Reams. October 2026. [https://discord.com/channels/960643023006490684/1243312334907375676/1565879387927224474 Discord message]</ref>
|-
|BBBB(3)
|≥ 127
|{{TM|1RB1LB_0RC0LA_1LC0LA|non_halt}} (beep: A0, boop: B1)
|Holdouts: 935.<ref name="calr3-bbbb">Charlie Reams, GitHub https://github.com/calr3/bbbb</ref>
|-
|BBBB(2, 3)
|≥ 187
|{{TM|2RB2LA1RA_1LB1LA0RB|non_halt}} (beep: B0, boop: B2)
|Holdouts: >2000.<ref name="calr3-bbbb"></ref>
|}

==Higher-order Beavers==
For higher order functions, using https://katelyndoucette.com/articles/building-the-busy-beaver-ladder definition, we can get BBBBB(1) = 2 and BBBBB(2) ≥ 18.

== References ==

[[category:Functions]]

Beeping Busy Beaver

2026-10-31T11:26:40Z

Frogmatic: Explain more of the significance of BBB in terms of simulation power

The '''Beeping Busy Beaver''' (BBB) function is a variant of the [[Busy Beaver Functions|Busy Beaver function]] proposed by Scott Aaronson in his 2020 [[Busy Beaver Frontier]] survey.<ref name=":0">Scott Aaronson. "The Busy Beaver Frontier". https://www.scottaaronson.com/papers/bb.pdf</ref> It is notable because it is uncomputable even if you have access to a halting oracle for Turing Machines. These can simulate problems in Π₂ and Σ₂ of the [[wikipedia:Arithmetic hierarchy|arithmetic hierarchy]], such as the [[wikipedia:Collatz conjecture|Collatz conjecture]] and [[wikipedia:Twin prime#Twin prime conjecture|twin prime conjecture]].

== Definition ==
Consider a ''beeping Turing machine'', which is a [[Turing machine]] that has a special state named "beep state". Every time the TM enters the "beep state" it ''beeps''. There are two possibilities, either this TM beeps a finite number of times (and thus there is a final beep) or it never stops beeping. Nick Drozd coined the term [[Quasihalt|quasihalting]] to describe the event when a TM last beeps. A TM quasihalts if it beeps only a finite number of times.<ref>Nick Drozd. 2020. [https://nickdrozd.github.io/2020/08/13/beeping-busy-beavers.html Beeping Busy Beavers].</ref> The Beeping Busy Beaver problem is analogous to the Busy Beaver problem, replacing halting with quasihalting. In other words, let <math>b(M)</math> be the number of steps the machine M takes until it quasihalts (beeps for the last time) if it quasihalts (we will say <math>b(M) = \infty</math> if the TM never stops beeping). Then

<math display="block">\operatorname{BBB}(n) := \max_{{M \in T(n)\ :\ b(M) < \infty}} b(M)</math>

where <math>T(n)</math> is the set of Turing machines with <math>n</math> states and two symbols.

Note that quasihalting does not require halting, so we can consider Turing machines which have no halt transitions. Thus the [[Tree Normal Form]] algorithm needs to be modified (to allow TMs with no halt transitions) when searching for BBB champions.

Another way to define BBB is to have Turing machines have a special beep transition instead of an entire beep state.

== Growth rate ==

It is easy to see that <math>\operatorname{BBB}(n) \ge \operatorname{BB}(n)</math>, by letting the beep state be the state that is reached immediately before the halt state.

In fact, BBB grows much faster than BB. BBB eventually dominates any computable function augmented with an oracle for computing BB. So, for example, there exists some N such that for all n > N:

* <math>BBB(n) > BB(BB(n))</math>
* <math>BBB(n) > BB^n(n)</math>, where <math>BB^k</math> represents k iterations of BB
* <math>BBB(n) > BB^{BB(n)}(n)</math>

etc.

== Champions ==
This table considers beep states.
{| class="wikitable"
|+
!Domain
!Max Steps
!Champions
|-
|BBB(1)
|= 1<ref name=":0" />
|{{TM|1RZ---|halt}}
|-
|BBB(2)
|= 6<ref name=":0" />
|{{TM|1RB1LB_1LB1LA|non_halt}}
|-
|BBB(3)
|= 55
|{{TM|1RB0LB_1LA0RC_1LC1LA|non_halt}}
|-
|BBB(4)
|= 32,779,478<ref name=":Coq-BBB4">[https://github.com/carrino/Coq-BBB4 Coq-BBB4]</ref>
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD|non_halt}}
|-
|BBB(5)*
|> 10<sup>14,006</sup><ref>Nick Drozd. https://scottaaronson.blog/?p=8088#comment-1981333</ref>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA|non_halt}}
|-
|BBB(2,3)
|59<ref>Nick Drozd, "[https://nickdrozd.github.io/2025/03/24/bbb-3-3.html BBB(3,3) > 10↑↑6]". Blog post. Accessed 15 August 2025.</ref>
|{{TM|1RB2LB1LA_2LB2RA0RA|non_halt}}
|-
|BBB(3,3)
|> 10 ↑↑ 6.74<ref name=":Nick Drozd Tetrational Blank26">Nick Drozd, "[https://nickdrozd.github.io/2026/09/01/tetrational-self-cleaning-turing-machine.html A Tetrational Self-Cleaning Turing Machine]". Blog post. Accessed 17 October 2026.</ref>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC|non_halt}}
|-
|BBB(2,4)
|> <math>2 \times 10^{23}</math><ref>Nick Drozd. "[https://nickdrozd.github.io/2022/02/11/latest-beeping-busy-beaver-results.html Latest Beeping Busy Beaver Results]". Accessed 15 August 2025.</ref>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA|non_halt}}
|}
<nowiki>*</nowiki> probably <math>BBB(5) > 10^{10^{10^5}}</math> due to four "[[probviously]]" quasihalting [[Cryptids]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/04/03/mother-of-giants.html Mother of Giants].</ref>

All known champions quasihalt by becoming [[Translated Cycler|Translated Cyclers]], a property which is known to be weaker than the general quasihalting condition.

==Beeping Booping Busy Beavers==
A further extension devised by Bram Cohen is as follows: a Turing machine has two special transitions, a beep transition and a boop transition, both of which may repeat infinitely often. The machine "outputs" an integer sequence, indicated by the number of beeps between successive boops. If any such integer is repeated an infinite number of times, the machine is said to '''pseudoquasihalt'''. Now if we define <math>g(M)</math> as the number of transitions for machine M to finish outputting that infinitely-repeated value for the first time (or ∞ if there is no such value), the Beeping Booping Busy Beaver number is:

<math display="block">\operatorname{BBBB}(n) := \max_{{M \in T(n)\ :\ g(M) < \infty}} g(M)</math>

These machines are equivalent to Turing machines with second-order oracles, and so their pseudoquasihalting behavior can describe sets in Π₃ and Σ₃ of the [[wikipedia:Arithmetic hierarchy|arithmetic hierarchy]].<ref>Bram Cohen. 2023. [https://bramcohen.com/p/beeping-booping-busy-beavers Beeping Booping Busy Beavers].</ref>

There are multiple models to choose from, depending on whether one allows multiple beep transitions and multiple boop transitions, or only one of each, or only one boop transition while making every transition a beep transition. All of these models have the same computational strength and corresponding growth rate, but each has its own advantages. The first is the most general, including all machines from the other two, and in that sense it is more natural, but that makes the space of candidate machines to search through significantly larger. The second is the original definition and it is relatively simple. The third is equivalent to simply considering the amount of time between successive boops, instead of the number of beeps between successive boops, and has the smallest space of candidate machines. It is not yet decided which of these models (or perhaps a model different from all three) should be chosen for "the" BBBB function.

It is known that for all three of these models, BBBB(1) = 2 and BBBB(2) = 18, with champion {{TM|0RB1LB_1LA1RB|non_halt}}.<ref>Charlie Reams. 2026. [https://discord.com/channels/960643023006490684/1243312334907375676/1565879387927224474 Discord message]</ref>

==Higher-order Beavers==
For higher order functions, using https://katelyndoucette.com/articles/building-the-busy-beaver-ladder definition, we can get BBBBB(1) = 2 and BBBBB(2) ≥ 18.

== References ==

[[category:Functions]]

1RB1LD 1RC1RB 1LC1LA 0RC0RD

2026-10-31T11:22:00Z

Frogmatic: Create article for the BBBB(2) champion

{{machine|1RB1LD_1RC1RB_1LC1LA_0RC0RD}}
{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD}} is the [[Beeping Booping Busy Beaver]] champion for BBBB(2), pseudoquasihalting after 18 steps. It was discovered and reported by Charlie Reams in October 2026, correcting an [[wikipedia:Off-by-one error|off-by-1 error]] in previous work.<ref>[https://discord.com/channels/960643023006490684/1243312334907375676/1565879387927224474 Discord message]</ref> This is proven to be the champion for this state and symbol count.<ref>https://github.com/calr3/bbbb</ref>

With boop transition A1 and beep transition A0, the machine boops at steps 6, 13, 18, ...; the first interval contains a single beep (therefore emitting a 1), and all later intervals contain no beeps (therefore emitting 0 infinitely thereafter). So, in the BBBB interpretation, it outputs an infinite stream of zeroes beyond this point and the first zero is emitted at transition 18, so BBBB(M) = 18.

== References ==

Beeping Booping Busy Beaver

2026-10-31T11:19:20Z

Frogmatic: Redirected page to Beeping Busy Beaver#Beeping Booping Busy Beavers

#REDIRECT [[Beeping Busy Beaver#Beeping Booping Busy Beavers]]

Cryptids

2026-10-31T09:29:52Z

Frogmatic: /* Cryptids for other BB-style functions */ Split behavior out from other notes for BBB, consistent with the BB table.

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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". The best known bounds for each problem (with respect to BB(...), BB(..., n), and [[BBi]](...)) are shown in bold.

{| class="wikitable"
|-
! Name !! Behaviour !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Source !! Date !! Discoverer !! Note
|-
|rowspan="2"| '''[[Logical independence|ZFC consistency]]'''
|rowspan="2"| Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is inconsistent.
|BB(432)
|
|[https://codeberg.org/ajwade/turing_machine_explorer/src/branch/main/zf2.tm Wade's machine]
|[https://codeberg.org/ajwade/turing_machine_explorer ajwade on codeberg.org]
|style="white-space: nowrap; text-align: right;"|Aug 2025
|Wade, based on work by CatIsFluffy and O'Rear
|This [[Logical independence#Axiom_of_Choice|also covers ZFC]].
|-
|'''BB(215)'''
|'''BBi(430)'''
|[https://github.com/VladimirReshetnikov/ProveIt/blob/main/Computability/BusyBeaver/Research/ZFC/k225-formal/lean215/k215_k230labels.tm Machine code]
|[https://github.com/VladimirReshetnikov/ProveIt/tree/main/Computability/BusyBeaver/Research/ZFC/k225-formal/verification/k215 VladimirReshetnikov on GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[wikipedia:Cleo (mathematician)|Vladimir Reshetnikov]]
|Machine-checked in Peano-Arithmetic by [https://github.com/elliotglazer/k215-pa elliotglazer Elliot Glazer].
|-
|'''Peano Arithmetic consistency'''
|Halts iff [[wikipedia:Peano arithmetic|Peano arithmetic]] is inconsistent.
|'''BB(372)'''
|'''BBi(742)'''
|
|[https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py LegionMammal978 GitHub]
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | '''Riemann Hypothesis'''
| rowspan="3" | Halts iff the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|BB(744)
|
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.tm Machine code]
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql metamath-turing-machines GitHub]
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|
|-
|BB(120)
|BBi(238)
|[https://arxiv.org/html/2609.30306v1#A9 Machine code]
|[https://arxiv.org/abs/2609.30306 arxiv]
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|
|-
|'''BB(95)'''
|'''BBi(189)'''
|[https://github.com/google-deepmind/cryptids/blob/main/artifacts/riemann_95/machine.txt Machine code]
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95/ DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"|'''Goldbach'''
| rowspan="2"|Halts iff [[wikipedia:Goldbach's conjecture|Goldbach's conjecture]] is false
|BB(25)
|BBi(49)
|{{WrapTM|Machine code|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|undecided}}
|[https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6 GitHub]
|style="white-space: nowrap; text-align: right;"|May 2016
|anonymous
|Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
|'''BB(21)'''
|'''BBi(41)'''
|{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21 DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| '''Fermat prime'''
| Halts if and only if there is a Fermat prime greater than 65537.
| '''BB(25)'''
| '''BBi(44)'''
|{{WrapTM|Machine code|1LB---_1LC---_1LD---_1LE---_1LF---_1LG0LO_0LY0RM_0LF0LI_0LJ1LI_1RK1LJ_0LL0RM_0RG1LL_0RN1RM_1RH1RN_1LP1LO_1RY0LQ_0LR0LQ_0RS1RT_1RS1LY_1RT1RU_0RV0RH_0RV0RW_0LZ1LX_0LX1LY_1LT1LY|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/fermatprimes calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| rowspan="2"| '''Erdős Ternary Conjecture'''
| rowspan="2"| Halts iff [https://www.erdosproblems.com/406 Erdős Problem 406] is false.
| BB(15) <br> '''BB(5,4)'''
| '''BBi(19)'''
|{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
{{WrapTM|BB(5,4) and BBi(19) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|[https://docs.bbchallenge.org/other/powers_of_two_15_2.txt bbchallenge], [https://docs.bbchallenge.org/other/powers_of_two_5_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
| [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods
| Erdős #406 is the conjecture that "For all n > 8, there is at least one 2 in the base-3 representation of 2^n." [https://arxiv.org/abs/2107.12475 arxiv preprint]
|-
| '''BB(14)'''
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
| [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|Not independently verified.
|-
| '''Erdős Ternary Conjecture''' variant
| Halts iff the following conjecture (a variant of #406) is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1."
| '''BB(4,4)'''
| '''BBi(15)'''
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
| [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
| Not independently verified.
|-
| '''Sloane Ternary Conjecture'''
| Halts if and only if the following [http://neilsloane.com/doc/persistence.html conjecture by N. J. A. Sloane] is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n."
| '''BB(6,4)'''
| '''BBi(21)'''
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
| Not independently verified.
|-
| '''Ternary Conjecture''' variant
| Halts if and only if a n such that 17*2^n has no digit 2 in base-3 exists.
| '''BB(15)'''
| '''BBi(29)'''
|{{WrapTM|Machine code|1RB1LB_1LA1RN_0RE1RD_0RC1RC_0LH1RC_0LG1LH_---1LF_1LF0LL_0LJ1LK_0RD1LI_1LI0LL_1LM1LN_0LI0LI_0RO1LL_1LL0LO|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|
|-
| rowspan="2"| '''Weak Collatz'''
| rowspan="2"|Halts iff the weak [[wikipedia:Collatz conjecture|Collatz conjecture]] is false.
|BB(124) <br> BB(43,4)
|BBi(121)
|
|[https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt bbchallenge], [https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). This is a weakened version of the usual Collatz conjecture, since it does not exclude the possibility of integers that are not part of any cycle at all. Not independently verified.
|-
|'''BB(14)''' <br> '''BB(10,3)'''
|'''BBi(26)'''
|style="word-break:break-word"|{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) and BBi(26) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/weakcollatz calr3 on GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| '''Odd multiply perfect'''
| Halts iff an odd [[wikipedia:Multiply perfect number|multiply perfect]] greater than 1 exists.
| '''BB(23,3)'''
| '''BBi(63)'''
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|[https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|Not independently verified.
|-
| '''Odd perfect'''
| Halts iff an odd [[wikipedia:Perfect number|perfect]] exists.
| '''BB(37)'''
| '''BBi(71)'''
|[https://github.com/calr3/tinycryptids/blob/main/oddperfect/machines/oddperfect_37.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/oddperfect calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Sierpiński'''
|Halts iff 78,557 is the smallest [[wikipedia:Sierpiński number|Sierpiński number]].
| '''BB(60)'''
| '''BBi(104)'''
|[https://github.com/calr3/tinycryptids/blob/main/sierpinski/machines/sierpinski_60.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/sierpinski/ calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Riesel'''
|Halts iff 509,203 is the smallest [[wikipedia:Riesel number|Riesel number]].
| '''BB(68)'''
| '''BBi(112)'''
|[https://github.com/loader3229/tinycryptids/blob/main/riesel/machines/riesel_min.txt Machine code]
|[https://github.com/loader3229/tinycryptids/tree/main/riesel loader3229 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]] and DeepSeek
|Includes Lean formalization. Not independently verified.
|-
| rowspan="2"| '''Fermat's Last Theorem'''
| rowspan="2"|Does not halt.
|BB(400)
|
| Not provided.
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994; it is now known that it cannot halt. 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.
|-
| '''BB(90)'''
| '''BBi(160)'''
|[https://github.com/calr3/tinycryptids/blob/main/fermat/machines/fermat_90.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/fermat calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Bigfoot''' - compiled
| Halts iff [[Bigfoot]] halts.
| '''[[BB(7)]]'''
| '''BBi(13)'''
| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot comment]
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states].
|-
| '''Hydra''' - compiled
| Halts iff [[Hydra]] halts.
| '''BB(9)'''
| '''BBi(15)'''
|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}
|[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|}

== Cryptids for other BB-style functions ==

=== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement but not a Π<sub>1</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! Behaviour !! BBB domain !! Machine !! Source !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|Probviously quasihalting.
|[[Beeping Busy Beaver|BBB(5)]]
|{{WrapTM|Machine code|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|Found "in the wild".
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|
|-
|Fermat Primes
|Quasihalts iff there are only finitely many Fermat primes.
|[[Beeping Busy Beaver|BBB(16)]]
|{{WrapTM|Machine code (beep state: P)|1LO1LA_0LI0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RH1RG_1RB1RH_---0LJ_1LK1LJ_1RA0LL_0LM0LL_0RN1RO_1RN1LA_1RO1RP_0LA0RB|non_halt}}
|[https://github.com/calr3/tinycryptids/tree/main/fermatprimesinf calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Not independently verified.
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|Quasihalts iff Twin Prime conjecture is false.
|[[Beeping Busy Beaver|BBB(18)]]
|{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|[https://github.com/calr3/tinycryptids/tree/main/twinprimes calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Not independently verified.
|-
|Collatz
|Quasihalts iff the [[wikipedia:Collatz conjecture|Collatz conjecture]] is false.
|[[Beeping Busy Beaver|BBB(17)]]
|{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|[https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md Mira-Cult GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|Not independently verified.
|}

=== BitChanger ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|}

=== Boolfuck ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|}

=== CounterScript ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|}

=== Fractran ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|}

=== Lambda Calculus ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|}

=== Pebble Automaton ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|}

=== Post Tag System ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|}

=== Register machine ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(130)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|}

=== Wang Tiles ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Beeping Busy Beaver

2026-10-31T08:48:46Z

Frogmatic: Correct value for BBBB(2)

The '''Beeping Busy Beaver''' (BBB) function is a variant of the [[Busy Beaver Functions|Busy Beaver function]] proposed by Scott Aaronson in his 2020 [[Busy Beaver Frontier]] survey.<ref name=":0">Scott Aaronson. "The Busy Beaver Frontier". https://www.scottaaronson.com/papers/bb.pdf</ref> It is notable because it is uncomputable even if you have access to a halting oracle for Turing Machines.

== Definition ==
Consider a ''beeping Turing machine'', which is a [[Turing machine]] that has a special state named "beep state". Every time the TM enters the "beep state" it ''beeps''. There are two possibilities, either this TM beeps a finite number of times (and thus there is a final beep) or it never stops beeping. Nick Drozd coined the term [[Quasihalt|quasihalting]] to describe the event when a TM last beeps. A TM quasihalts if it beeps only a finite number of times.<ref>Nick Drozd. 2020. [https://nickdrozd.github.io/2020/08/13/beeping-busy-beavers.html Beeping Busy Beavers].</ref> The Beeping Busy Beaver problem is analogous to the Busy Beaver problem, replacing halting with quasihalting. In other words, let <math>b(M)</math> be the number of steps the machine M takes until it quasihalts (beeps for the last time) if it quasihalts (we will say <math>b(M) = \infty</math> if the TM never stops beeping). Then

<math display="block">\operatorname{BBB}(n) := \max_{{M \in T(n)\ :\ b(M) < \infty}} b(M)</math>

where <math>T(n)</math> is the set of Turing machines with <math>n</math> states and two symbols.

Note that quasihalting does not require halting, so we can consider Turing machines which have no halt transitions. Thus the [[Tree Normal Form]] algorithm needs to be modified (to allow TMs with no halt transitions) when searching for BBB champions.

Another way to define BBB is to have Turing machines have a special beep transition instead of an entire beep state.

== Growth rate ==

It is easy to see that <math>\operatorname{BBB}(n) \ge \operatorname{BB}(n)</math>, by letting the beep state be the state that is reached immediately before the halt state.

In fact, BBB grows much faster than BB. BBB eventually dominates any computable function augmented with an oracle for computing BB. So, for example, there exists some N such that for all n > N:

* <math>BBB(n) > BB(BB(n))</math>
* <math>BBB(n) > BB^n(n)</math>, where <math>BB^k</math> represents k iterations of BB
* <math>BBB(n) > BB^{BB(n)}(n)</math>

etc.

== Champions ==
This table considers beep states.
{| class="wikitable"
|+
!Domain
!Max Steps
!Champions
|-
|BBB(1)
|= 1<ref name=":0" />
|{{TM|1RZ---|halt}}
|-
|BBB(2)
|= 6<ref name=":0" />
|{{TM|1RB1LB_1LB1LA|non_halt}}
|-
|BBB(3)
|= 55
|{{TM|1RB0LB_1LA0RC_1LC1LA|non_halt}}
|-
|BBB(4)
|= 32,779,478<ref name=":Coq-BBB4">[https://github.com/carrino/Coq-BBB4 Coq-BBB4]</ref>
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD|non_halt}}
|-
|BBB(5)*
|> 10<sup>14,006</sup><ref>Nick Drozd. https://scottaaronson.blog/?p=8088#comment-1981333</ref>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA|non_halt}}
|-
|BBB(2,3)
|59<ref>Nick Drozd, "[https://nickdrozd.github.io/2025/03/24/bbb-3-3.html BBB(3,3) > 10↑↑6]". Blog post. Accessed 15 August 2025.</ref>
|{{TM|1RB2LB1LA_2LB2RA0RA|non_halt}}
|-
|BBB(3,3)
|> 10 ↑↑ 6.74<ref name=":Nick Drozd Tetrational Blank26">Nick Drozd, "[https://nickdrozd.github.io/2026/09/01/tetrational-self-cleaning-turing-machine.html A Tetrational Self-Cleaning Turing Machine]". Blog post. Accessed 17 October 2026.</ref>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC|non_halt}}
|-
|BBB(2,4)
|> <math>2 \times 10^{23}</math><ref>Nick Drozd. "[https://nickdrozd.github.io/2022/02/11/latest-beeping-busy-beaver-results.html Latest Beeping Busy Beaver Results]". Accessed 15 August 2025.</ref>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA|non_halt}}
|}
<nowiki>*</nowiki> probably <math>BBB(5) > 10^{10^{10^5}}</math> due to four "[[probviously]]" quasihalting [[Cryptids]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/04/03/mother-of-giants.html Mother of Giants].</ref>

All known champions quasihalt by becoming [[Translated Cycler|Translated Cyclers]], a property which is known to be weaker than the general quasihalting condition.

==Beeping Booping Busy Beavers==
A further extension devised by Bram Cohen is as follows: a Turing machine has two special transitions, a beep transition and a boop transition, both of which may repeat infinitely often. The machine "outputs" an integer sequence, indicated by the number of beeps between successive boops. If any such integer is repeated an infinite number of times, the machine is said to '''pseudoquasihalt'''. Now if we define <math>g(M)</math> as the number of transitions for machine M to finish outputting that infinitely-repeated value for the first time (or ∞ if there is no such value), the Beeping Booping Busy Beaver number is:

<math display="block">\operatorname{BBBB}(n) := \max_{{M \in T(n)\ :\ g(M) < \infty}} g(M)</math>

These machines are equivalent to Turing machines with second-order oracles, and so their pseudoquasihalting behavior can describe sets in Π₃ and Σ₃ of the [[wikipedia:Arithmetic hierarchy|arithmetic hierarchy]].<ref>Bram Cohen. 2023. [https://bramcohen.com/p/beeping-booping-busy-beavers Beeping Booping Busy Beavers].</ref>

There are multiple models to choose from, depending on whether one allows multiple beep transitions and multiple boop transitions, or only one of each, or only one boop transition while making every transition a beep transition. All of these models have the same computational strength and corresponding growth rate, but each has its own advantages. The first is the most general, including all machines from the other two, and in that sense it is more natural, but that makes the space of candidate machines to search through significantly larger. The second is the original definition and it is relatively simple. The third is equivalent to simply considering the amount of time between successive boops, instead of the number of beeps between successive boops, and has the smallest space of candidate machines. It is not yet decided which of these models (or perhaps a model different from all three) should be chosen for "the" BBBB function.

It is known that for all three of these models, BBBB(1) = 2 and BBBB(2) = 18, with champion {{TM|0RB1LB_1LA1RB|non_halt}}.<ref>Charlie Reams. 2026. [https://discord.com/channels/960643023006490684/1243312334907375676/1565879387927224474 Discord message]</ref>

==Higher-order Beavers==
For higher order functions, using https://katelyndoucette.com/articles/building-the-busy-beaver-ladder definition, we can get BBBBB(1) = 2 and BBBBB(2) ≥ 18.

== References ==

[[category:Functions]]

Champions

2026-10-28T21:01:39Z

Frogmatic: /* 2-Symbol TMs */ Use consistent style for missing champions (at least in the dense part of the table)

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]]; see [[measuring large numbers]] for more information. Machines below 6 states are listed in [[Tree Normal Form|TNF]] format, where machines which tie for champion are all listed. Note that most champions above 6 states are self-reported and have not been independently verified.



{| class="wikitable"
|+
!Domain
!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>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8</math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|mxdys in 2025
|See mxdys's analysis on the TM page
|-
|[[BB(7)]]
|<math>> 3 \uparrow^{20} 3 > f_\omega(20)</math>
|{{TM|1RB0RG_1RC1RB_0RD1RZ_0LE0LD_0LF1LE_1RA1LF_1LC0RB|halt}}
|Jeff Ketchersid and Gemini in 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB7/1RB0RG_1RC1RB_0RD---_0LE0LD_0LF1LE_1RA1LF_0LD0RB.md Claimed analysis by Claude] never verified.
|-
|[[BB(8)]]
|<math>> f_\omega^3(12)</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG1RZ|halt}}
|Jeff Ketchersid and Gemini in 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB8/1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LH0LC_1RG1LD_0RG0RF_1RG---.md Claimed analysis by Claude] never verified.
|-
|BB(9)
|<math>> f_\omega^7(12)</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI1RZ_1RI1RC|halt}}
|Jeff Ketchersid and Gemini in 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/blob/main/BB9/1RB0RA_1LC1LF_1RD0LB_1RA1LE_0LH0LC_1RG1LD_0RG0RF_0LI---_1RI1RC.md Claimed analysis by Claude] never verified.
|-
|BB(10)
|<math>> f_{\omega+1}(2 \uparrow^{36} 3) > g_{64}</math>
|<span style="word-break:break-all">{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1LI0LC_0RG1LD_1RH1LG_1LC0RG_0LJ0LI_1RZ0LC|halt}}</span>
|[https://discord.com/channels/960643023006490684/1331570843829932063/1557806386073702622 Jeff Ketchersid and Claude in 2026 (clamied)]
|[https://github.com/jeff87654/busy-beaver/tree/main/BB10 Claimed analysis by Claude] never verified.
|-
|BB(11)
|<math>> f_{\omega+1}(f_\omega^2(5))</math>
|<span style="word-break:break-all">{{TM|1RG0LH_1RK1RC_0LD1LA_0RE1LD_0LI1RF_1RZ0RG_1RB1RG_1LA1LH_0LK0LJ_1LH1LI_1LJ1RE|halt}}</span>
|Jeff Ketchersid and Gemini in 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB11 Claimed analysis by Claude] never verified.
|-
|BB(12)
|<math>> f_{\omega+2}(f_{\omega+1}^2(3))</math>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK|halt}}</span>
|Jeff Ketchersid and Gemini in 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB12 Claimed analysis by Claude] never verified.
|-
|...
|
|
|
|
|-
|BB(14)
|<math>> f_{\omega+2}^3(15)</math>
|<span style="word-break:break-all">{{TM|1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LN1RF_1LJ0RA_1RA0LH_1LG1LH_1RK0LM_1RZ0LE_0LE0RJ_1LH1LM_0LH0LL_0LG1LN|halt}}</span>
|Jeff Ketchersid and Gemini in 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB14 Claimed analysis by Claude] never verified.
|-
|...
|
|
|
|
|-
|BB(20)
|<math>> f_{\omega + 6}^2(4)</math>
|<span style="word-break:break-all">{{TM|1RR---_0LJ0RC_1RB1LD_0LD1LE_1LF1RG_1LI0LE_0RE1LD_1RB1RH_1RH1RN_1LK1LJ_0LL1LJ_1RM1RA_0RM0RH_0LO1RN_1RH0LP_1RH1LQ_1RH1LO_1LS1RR_1RZ1LT_0LH0LS|halt}}</span>
|Jacobzheng in 2026
|
|-
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|<span style="word-break:break-all">{{TM|0LI0LF_0RJ1RG_0RD1LC_1RH1RE_1LO0RH_1LA1LF_0LC1LA_1RB1RH_1RD0LF_1LP0RK_0LM1LL_0LL1LM_1LE1LN_0RQ0LM_0RP1LO_1LR0RH_1LF1RQ_1RZ0LS_1LH0LT_1LH1LU_0LE1LR|halt}}</span>
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|
|-
|BB(29)
|<math>> f_{\omega^\omega}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB29/hydra29.md]
|mxdys in 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB29 Claimed analysis by Claude] never verified.
|-
|BB(31)
|<math>> f_{\omega \uparrow \uparrow 4}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB31/hydra31.md]
|mxdys in 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB31 Claimed analysis by Claude] never verified.
|-
|BB(33)
|<math>> f_{\omega \uparrow \uparrow 6}(3)</math>
|[https://github.com/jeff87654/busy-beaver/blob/main/BB33/hydra33.md]
|mxdys in 2026 (claimed)
|[https://github.com/jeff87654/busy-beaver/tree/main/BB33 Claimed analysis by Claude] never verified.
|-
|BB(51)
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|[[Talk:Champions|See talk page]]
|[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>
|
|-
|BB(1015)
|<math>> D^{5}(99)</math> = Loader's Number
|
|CatsAreFluffy in 2024<ref>https://github.com/CatsAreFluffy/metamath-turing-machines/commit/85948b04fc4aeb983ca6d63d6aee5ad6ef308bfe</ref>
|
|}

=== Conjectured ===

{| class="wikitable"
|+
!Domain
!Runtime
!Champions
!Discovered by
!Notes
|-
|BB(67)
|<math>> q^{81}(q(5)-2)</math>
|[https://wiki.bbchallenge.org/wiki/Talk:Champions]
|[https://discord.com/channels/960643023006490684/1331570843829932063/1339669138510970881 racheline in 2025]
|<math>q(n)</math> is derived from [https://en.wikipedia.org/wiki/Laver%20table Laver tables]. From [https://discord.com/channels/960643023006490684/960643023530762341/1525489299032641657 racheline]: "The known information about q(n) isn't enough to prove that this machine scores higher than the shown champions with fewer states, or that it scores lower than the shown champions with more states."
|}

== 3-Symbol TMs ==

{| class="wikitable"
|+
!Domain
!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
|
|-
|BB(5,3)
|
|
|
|
|-
|BB(6,3)
|<math>> f_\omega^2(10\uparrow\uparrow\uparrow 6)</math>
|{{TM|1LE2LD2LC_2RB1RB1RA_2RB1LC0LC_1RB1LD2LD_2RD2LE2LF_1RZ---1LE|halt}}
|Jacobzheng in 2026
|
|-
|BB(7,3)
|
|
|
|
|-
|BB(8,3)
|<math>> f_{\omega+1}(f_\omega(f_7(2))) > g_{64}</math>
|<span style="word-break:break-all">{{TM|1LD1LE2RA_1RE2LE2RB_1RD2RA2LC_1RB1RD2RB_1LH2RF1LC_1RZ0LF1LG_1RD2LG2LG_0LD1LH1LH|halt}}</span>
|Jacobzheng in 2026
|
|-
|BB(9,3)
|
|
|
|
|-
|BB(10,3)
|
|
|
|
|-
|BB(11,3)
|<math>> f_{\omega+3}^2(4)</math>
|<span style="word-break:break-all">{{TM|1LD2RG2LA_2LE2LH2RB_2LE1RC2RB_0RC1RD0LG_0LF1LE1LE_1RG1RI---_0RC1LH2RG_1RC2LH1LA_1RI1RI1LJ_0LA2LK---_1RZ2LJ---|halt}}</span>
|Jacobzheng in 2026
|
|-
|...
|
|
|
|
|-
|BB(36,3)
|<math>> f_{\varepsilon_0}(7)</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(36,3)</ref>
|
|-
|BB(37,3)
|<math>> f_{\varepsilon_0}(373\,676\,378)</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(37,3)</ref>
|
|}

== 4-Symbol TMs ==
{| class="wikitable"
|+
!Domain
!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]
|-
|[[BB(4,4)]]
|<math>> 2 \uparrow^{509} 3 > f_\omega(508)</math>
|{{TM|0LB3LB1RZ2RA_1LD3RB1LC2RA_3RB1LB3LC2RC_0RD2RC2LB0RA|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|BB(5,4)
|<math>> f_\omega^2(10\uparrow^4 10\uparrow\uparrow 65\,533)</math>
|<span style="word-break:break-all">{{TM|1LD2RA3RB1LC_1LD2LA3RB---_2RA1LA3RC2LC_2RA---2LD2LE_------1RZ1LD|halt}}</span>
|Jacobzheng in 2026
|
|-
|BB(6,4)
|<math>> f_{\omega+1}(97) > g_{64}</math>
|<span style="word-break:break-all">{{TM|0RC3LB2RD2RA_1LF3RB1LC2RA_3RB1LB3LC2RC_1RZ2LD1LD2LE_3RE3LB2RE2LE_2RB2LF0LB2RF|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(31,4)
|<math>> f_{\varepsilon_0}(373\,676\,378)</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(31,4)</ref>
|
|}

== 5-Symbol TMs ==
{| class="wikitable"
!Domain
!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
|
|-
|BB(4,5)
|<math>> f_{\omega+1}(27\,041) > g_{64}</math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA4RD_3RB1LB3LC2RC4LC_2RC2RD1RZ4RB1LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(19,5)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<span style="word-break:break-all">{{TM|1RB2LA1RA2LB2LA_0LA2RB3RB4RA1LQ_1RZ1RC2LD------_0LK3RD2RD2RD4RD_------0RH1LS0RF_1LG1RF2RF---4RF_4LE1LG2LG---4LG_0LI1RH2RH---4RH_0RK0LJ------4RK_2LE1LJ2LJ---4LJ_4RP---0LE---0LK_2RQ---2LL---4LL_2LN---2RM3RM4RM_4RQ2RQ2LN3LN4LN_4LN---2RO3RO4RO_3LL1LP2LP1LQ4LP_1RC3RR1RM3LP0RO_3LL1RR2RR0RR4RR_1RC1LS2LS1LS4LS|halt}}</span>
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(19,5)</ref>
|-
|BB(94,5)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|
|}

== 6-Symbol TMs ==
{| class="wikitable"
!Domain
!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]
|-
|BB(3,6)
|<math>> f_\omega^3(15)</math>
|{{TM|1RB3LB4LC2RA4LB5LB_2LC3RB1LC2RA5LC1RZ_3RB1LB3LC2RC4LC5LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(61,6)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|
|}

== 7-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|[[BB(2,7)]]
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
|{{TM|1RB3RB4LA1LB5LA6LB5RA_2LA6RA4RB1RZ3LB6LA4RB|halt}}
|[https://discord.com/channels/960643023006490684/960643023530762341/1563057558011252761 roromachine14 in 2026]
|
|-
|BB(3,7)
|<math>> f_\omega^4(15)</math>
|{{TM|1RB3LB4LC2RA4LB5LB6LB_2LC3RB1LC2RA5LC6LC1RZ_3RB1LB3LC2RC4LC5LC6LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(18,7)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<span style="word-break:break-all">{{TM|1RB2LA1RA2LB2LA------_0LA2RB3RB4RA1LP------_0RZ1RC2LD------------_0LJ3RD2RD2RD4RD------_4LE1LE0RG1LR0RF2LE4LE_1LE1RF5RF---6RF------_0LH1RG2RG---4RG------_0RJ0LI------4RJ------_2LE1LI2LI---4LI------_4RO---0LE---0LJ------_2RP---2LK---4LK------_2LM---2RL3RL4RL------_4RP2RP2LM3LM4LM------_4LM---2RN3RN4RN------_3LK1LO2LO1LP4LO------_1RC3RQ1RL3LO0RN------_3LK1RQ2RQ0RQ4RQ------_1RC1LR2LR1LR4LR------|halt}}</span>
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(18,7)</ref>
|-
|BB(46,7)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|
|-
|BB(134,7)
|<math>> f_{\psi_0(\Omega_\omega)}(2\,050)</math>
|
|LittlePeng9+Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/Analysis_of_LittlePeng9%27s_work/Hydra_machines</ref>
|
|}

== 8-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|BB(2,8)
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
|{{TM|1RB3RB4LA1LB5LA6LB5RA1RZ_2LA6RA4RB7RA3LB6LA4RB4RA|halt}}
|[https://discord.com/channels/960643023006490684/1084047886494470185/1563070360956698804 roromachine14 in 2026]
|
|-
|BB(3,8)
|<math>> f_\omega^5(15)</math>
|{{TM|1RB3LB4LC2RA4LB5LB6LB7LB_2LC3RB1LC2RA5LC6LC7LC1RZ_3RB1LB3LC2RC4LC5LC6LC7LC|halt}}
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(17,8)
|<math>> f_{\varepsilon_0}(1.7\cdot10^{352})</math>
|<span style="word-break:break-all">{{TM|1RB2LA1RA2LB2LA---------_0LA2RB3RB4RA1LO---------_0RZ1RC2LD---------------_0LJ3RD2RD2RD4RD---------_4LE1LE0RG1LQ0RF2LE4LE---_1LE1RF5RF---6RF---------_0LH1RG2RG---4RG---------_0RJ0LI------4RJ---------_2LE1LI2LI---4LI---------_4RN---0LE---0LJ---------_2RO2RO2LK3LK4LK2LK4LK4RO_2LK---5RL3RL6RL---------_4LK---5RM3RM6RM---------_3LK1LN2LN1LO4LN---------_1RC3RP1RL3LN7RM---------_3LK1RP2RP0RP4RP---------_1RC1LQ2LQ1LQ4LQ---------|halt}}</span>
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User:Wythagoras/Rado%27s_sigma_function/BB(17,8)</ref>
|
|}

== 9-Symbol TMs ==
{| class="wikitable"
!Domain
!Runtime
!Champions
!Discovered by
!Verification
|-
|BB(2,9)
|<math>> 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 155</math>
|{{TM|1RB3RB4LA1LB5LA6LB5RA8RA1RZ_2LA6RA4RB7RA3LB6LA4RB4RA4RA|halt}}
|[https://discord.com/channels/960643023006490684/1084047886494470185/1563070360956698804 roromachine14 in 2026]
|
|-
|BB(3,9)
|<math>> f_\omega^6(15)</math>
|<span style="word-break:break-all">{{TM|1RB3LB4LC2RA4LB5LB6LB7LB8LB_2LC3RB1LC2RA5LC6LC7LC8LC1RZ_3RB1LB3LC2RC4LC5LC6LC7LC8LC|halt}}</span>
|[[User:Loader3229]] in 2026
|Not independently verified.
|-
|...
|
|
|
|
|-
|BB(42,9)
|<math>> f_{\psi_0(\Omega_2)}(1.7\cdot10^{352})</math>
|
|Wythagoras in 2014<ref>https://googology.fandom.com/wiki/User_blog:Wythagoras/Hydra_machines</ref>
|
|}

== Zoology ==
There are many types of champions.

Note: the scale here is not formally defined. It mostly serves to estimate the average values produced by a type of champions.
{| class="wikitable"
|+
!Classification
!Description
!Examples
!Scale
|-
|Trivial
|The simplest champions that can exist. They mostly appear in some BB-adjacent functions like [[Fractran|BBf]] or [[CounterScript|BBCS]].
|
* {{TM|1RZ---|halt}}
|<math>O(n)</math>
|-
|Chaotic
|Have a chaotic behavior with repeating patterns that go back and forth.
|
* {{TM|1RB1LB_1LA1RZ|halt}}
* {{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
* {{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|<math>O(2^n)</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|1RB2LB1RZ_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_1LB1LA3RB1RZ|halt}}
* {{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|<math>O(2^n)</math>
|-
|Random walk
|Compute a [[Collatz-like]] function with a random walk.
|
|<math>O(2^{2^n})</math>
|-
|Tetrational
|
|
|<math>O(2 \uparrow\uparrow n)</math>
|-
|Ackermannian
|
|
|<math>O(2 \uparrow^n 3)</math>
|}

== References ==

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

Pseudoquasihalting

2026-10-28T19:52:06Z

Frogmatic: Redirect unusual terminology to the relevant parent article

#REDIRECT [[Beeping Busy Beaver#Beeping Booping Busy Beavers]]

Pseudoquasihalt

2026-10-28T19:51:34Z

Frogmatic: Redirect unusual terminology to the relevant parent article

#REDIRECT [[Beeping Busy Beaver#Beeping Booping Busy Beavers]]

Beeping Busy Beaver

2026-10-28T19:46:53Z

Frogmatic: /* Beeping Booping Busy Beavers */ Add some more explanation to BBBB's definition

The '''Beeping Busy Beaver''' (BBB) function is a variant of the [[Busy Beaver Functions|Busy Beaver function]] proposed by Scott Aaronson in his 2020 [[Busy Beaver Frontier]] survey.<ref name=":0">Scott Aaronson. "The Busy Beaver Frontier". https://www.scottaaronson.com/papers/bb.pdf</ref> It is notable because it is uncomputable even if you have access to a halting oracle for Turing Machines.

== Definition ==
Consider a ''beeping Turing machine'', which is a [[Turing machine]] that has a special state named "beep state". Every time the TM enters the "beep state" it ''beeps''. There are two possibilities, either this TM beeps a finite number of times (and thus there is a final beep) or it never stops beeping. Nick Drozd coined the term [[Quasihalt|quasihalting]] to describe the event when a TM last beeps. A TM quasihalts if it beeps only a finite number of times.<ref>Nick Drozd. 2020. [https://nickdrozd.github.io/2020/08/13/beeping-busy-beavers.html Beeping Busy Beavers].</ref> The Beeping Busy Beaver problem is analogous to the Busy Beaver problem, replacing halting with quasihalting. In other words, let <math>b(M)</math> be the number of steps the machine M takes until it quasihalts (beeps for the last time) if it quasihalts (we will say <math>b(M) = \infty</math> if the TM never stops beeping). Then

<math display="block">\operatorname{BBB}(n) := \max_{{M \in T(n)\ :\ b(M) < \infty}} b(M)</math>

where <math>T(n)</math> is the set of Turing machines with <math>n</math> states and two symbols.

Note that quasihalting does not require halting, so we can consider Turing machines which have no halt transitions. Thus the [[Tree Normal Form]] algorithm needs to be modified (to allow TMs with no halt transitions) when searching for BBB champions.

Another way to define BBB is to have Turing machines have a special beep transition instead of an entire beep state.

== Growth rate ==

It is easy to see that <math>\operatorname{BBB}(n) \ge \operatorname{BB}(n)</math>, by letting the beep state be the state that is reached immediately before the halt state.

In fact, BBB grows much faster than BB. BBB eventually dominates any computable function augmented with an oracle for computing BB. So, for example, there exists some N such that for all n > N:

* <math>BBB(n) > BB(BB(n))</math>
* <math>BBB(n) > BB^n(n)</math>, where <math>BB^k</math> represents k iterations of BB
* <math>BBB(n) > BB^{BB(n)}(n)</math>

etc.

== Champions ==
This table considers beep states.
{| class="wikitable"
|+
!Domain
!Max Steps
!Champions
|-
|BBB(1)
|= 1<ref name=":0" />
|{{TM|1RZ---|halt}}
|-
|BBB(2)
|= 6<ref name=":0" />
|{{TM|1RB1LB_1LB1LA|non_halt}}
|-
|BBB(3)
|= 55
|{{TM|1RB0LB_1LA0RC_1LC1LA|non_halt}}
|-
|BBB(4)
|= 32,779,478<ref name=":Coq-BBB4">[https://github.com/carrino/Coq-BBB4 Coq-BBB4]</ref>
|{{TM|1RB1LD_1RC1RB_1LC1LA_0RC0RD|non_halt}}
|-
|BBB(5)*
|> 10<sup>14,006</sup><ref>Nick Drozd. https://scottaaronson.blog/?p=8088#comment-1981333</ref>
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_0RC0LA|non_halt}}
|-
|BBB(2,3)
|59<ref>Nick Drozd, "[https://nickdrozd.github.io/2025/03/24/bbb-3-3.html BBB(3,3) > 10↑↑6]". Blog post. Accessed 15 August 2025.</ref>
|{{TM|1RB2LB1LA_2LB2RA0RA|non_halt}}
|-
|BBB(3,3)
|> 10 ↑↑ 6.74<ref name=":Nick Drozd Tetrational Blank26">Nick Drozd, "[https://nickdrozd.github.io/2026/09/01/tetrational-self-cleaning-turing-machine.html A Tetrational Self-Cleaning Turing Machine]". Blog post. Accessed 17 October 2026.</ref>
|{{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC|non_halt}}
|-
|BBB(2,4)
|> <math>2 \times 10^{23}</math><ref>Nick Drozd. "[https://nickdrozd.github.io/2022/02/11/latest-beeping-busy-beaver-results.html Latest Beeping Busy Beaver Results]". Accessed 15 August 2025.</ref>
|{{TM|1RB2LA1RA1LB_0LB2RB3RB1LA|non_halt}}
|}
<nowiki>*</nowiki> probably <math>BBB(5) > 10^{10^{10^5}}</math> due to four "[[probviously]]" quasihalting [[Cryptids]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/04/03/mother-of-giants.html Mother of Giants].</ref>

All known champions quasihalt by becoming [[Translated Cycler|Translated Cyclers]], a property which is known to be weaker than the general quasihalting condition.

==Beeping Booping Busy Beavers==
A further extension devised by Bram Cohen is as follows: a Turing machine has two special transitions, a beep transition and a boop transition, both of which may repeat infinitely often. The machine "outputs" an integer sequence, indicated by the number of beeps between successive boops. If any such integer is repeated an infinite number of times, the machine is said to '''pseudoquasihalt'''. Now if we define <math>g(M)</math> as the number of transitions for machine M to finish outputting that infinitely-repeated value for the first time (or ∞ if there is no such value), the Beeping Booping Busy Beaver number is:

<math display="block">\operatorname{BBBB}(n) := \max_{{M \in T(n)\ :\ g(M) < \infty}} g(M)</math>

These machines are equivalent to Turing machines with second-order oracles, and so their pseudoquasihalting behavior can describe sets in Π₃ and Σ₃ of the [[wikipedia:Arithmetic hierarchy|arithmetic hierarchy]].<ref>Bram Cohen. 2023. [https://bramcohen.com/p/beeping-booping-busy-beavers Beeping Booping Busy Beavers].</ref>

There are multiple models to choose from, depending on whether one allows multiple beep transitions and multiple boop transitions, or only one of each, or only one boop transition while making every transition a beep transition. All of these models have the same computational strength and corresponding growth rate, but each has its own advantages. The first is the most general, including all machines from the other two, and in that sense it is more natural, but that makes the space of candidate machines to search through significantly larger. The second is the original definition and it is relatively simple. The third is equivalent to simply considering the amount of time between successive boops, instead of the number of beeps between successive boops, and has the smallest space of candidate machines. It is not yet decided which of these models (or perhaps a model different from all three) should be chosen for "the" BBBB function.

It is known that for all three of these models, BBBB(1) = 2 and BBBB(2) = 17.<ref>[https://discord.com/channels/960643023006490684/1243312334907375676/1417836470793670697 Discord message]</ref>

==Higher-order Beavers==
For higher order functions, using https://katelyndoucette.com/articles/building-the-busy-beaver-ladder definition, we can get BBBBB(1) = 2 and BBBBB(2) ≥ 18.

== References ==

[[category:Functions]]

Cryptids

2026-10-27T19:48:27Z

Frogmatic: /* Larger Cryptids */ Remove the antihydra like -- this is not really a designed cryptid since it doesn't refer to any known problem. Could be created as its own page maybe?

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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". The best known bounds for each problem (with respect to BB(...), BB(..., n), and [[BBi]](...)) are shown in bold.

{| class="wikitable"
|-
! Name !! Behaviour !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Source !! Date !! Discoverer !! Note
|-
|rowspan="2"| '''[[Logical independence|ZF consistency]]'''
|rowspan="2"| Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|BB(432)
|
|[https://codeberg.org/ajwade/turing_machine_explorer/src/branch/main/zf2.tm Wade's machine]
|[https://codeberg.org/ajwade/turing_machine_explorer ajwade on codeberg.org]
|style="white-space: nowrap; text-align: right;"|Aug 2025
|Wade, based on work by CatIsFluffy and O'Rear
|This [[Logical independence#Axiom_of_Choice|also covers ZFC]].
|-
|'''BB(215)'''
|'''BBi(430)'''
|[https://github.com/VladimirReshetnikov/ProveIt/blob/main/Computability/BusyBeaver/Research/ZFC/k225-formal/lean215/k215_k230labels.tm Machine code]
|[https://github.com/VladimirReshetnikov/ProveIt/tree/main/Computability/BusyBeaver/Research/ZFC/k225-formal/verification/k215 VladimirReshetnikov on GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[wikipedia:Cleo (mathematician)|Vladimir Reshetnikov]]
|Machine-checked in Peano-Arithmetic by [https://github.com/elliotglazer/k215-pa elliotglazer Elliot Glazer].
|-
|'''Peano Arithmetic consistency'''
|Halts iff [[wikipedia:Peano arithmetic|Peano arithmetic]] is inconsistent.
|'''BB(372)'''
|'''BBi(742)'''
|
|[https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py LegionMammal978 GitHub]
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | '''Riemann Hypothesis'''
| rowspan="3" | Halts iff the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|BB(744)
|
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.tm Machine code]
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql metamath-turing-machines GitHub]
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|
|-
|BB(120)
|BBi(238)
|[https://arxiv.org/html/2609.30306v1#A9 Machine code]
|[https://arxiv.org/abs/2609.30306 arxiv]
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|
|-
|'''BB(95)'''
|'''BBi(189)'''
|[https://github.com/google-deepmind/cryptids/blob/main/artifacts/riemann_95/machine.txt Machine code]
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95/ DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"|'''Goldbach'''
| rowspan="2"|Halts iff [[wikipedia:Goldbach's conjecture|Goldbach's conjecture]] is false
|BB(25)
|BBi(49)
|{{WrapTM|Machine code|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|undecided}}
|[https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6 GitHub]
|style="white-space: nowrap; text-align: right;"|May 2016
|anonymous
|Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
|'''BB(21)'''
|'''BBi(41)'''
|{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21 DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| '''Fermat prime'''
| Halts if and only if there is a Fermat prime greater than 65537.
| '''BB(25)'''
| '''BBi(44)'''
|{{WrapTM|Machine code|1LB---_1LC---_1LD---_1LE---_1LF---_1LG0LO_0LY0RM_0LF0LI_0LJ1LI_1RK1LJ_0LL0RM_0RG1LL_0RN1RM_1RH1RN_1LP1LO_1RY0LQ_0LR0LQ_0RS1RT_1RS1LY_1RT1RU_0RV0RH_0RV0RW_0LZ1LX_0LX1LY_1LT1LY|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/fermatprimes calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| rowspan="2"| '''Erdős Ternary Conjecture'''
| rowspan="2"| Halts iff [https://www.erdosproblems.com/406 Erdős Problem 406] is false.
| BB(15) <br> '''BB(5,4)'''
| '''BBi(19)'''
|{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
{{WrapTM|BB(5,4) and BBi(19) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|[https://docs.bbchallenge.org/other/powers_of_two_15_2.txt bbchallenge], [https://docs.bbchallenge.org/other/powers_of_two_5_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
| [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods
| Erdős #406 is the conjecture that "For all n > 8, there is at least one 2 in the base-3 representation of 2^n." [https://arxiv.org/abs/2107.12475 arxiv preprint]
|-
| '''BB(14)'''
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
| [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|Not independently verified.
|-
| '''Erdős Ternary Conjecture''' variant
| Halts iff the following conjecture (a variant of #406) is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1."
| '''BB(4,4)'''
| '''BBi(15)'''
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
| [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
| Not independently verified.
|-
| '''Sloane Ternary Conjecture'''
| Halts if and only if the following [http://neilsloane.com/doc/persistence.html conjecture by N. J. A. Sloane] is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n."
| '''BB(6,4)'''
| '''BBi(21)'''
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
| Not independently verified.
|-
| '''Ternary Conjecture''' variant
| Halts if and only if a n such that 17*2^n has no digit 2 in base-3 exists.
| '''BB(15)'''
| '''BBi(29)'''
|{{WrapTM|Machine code|1RB1LB_1LA1RN_0RE1RD_0RC1RC_0LH1RC_0LG1LH_---1LF_1LF0LL_0LJ1LK_0RD1LI_1LI0LL_1LM1LN_0LI0LI_0RO1LL_1LL0LO|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|
|-
| rowspan="2"| '''Weak Collatz'''
| rowspan="2"|Halts iff the weak [[wikipedia:Collatz conjecture|Collatz conjecture]] is false.
|BB(124) <br> BB(43,4)
|BBi(121)
|
|[https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt bbchallenge], [https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). This is a weakened version of the usual Collatz conjecture, since it does not exclude the possibility of integers that are not part of any cycle at all. Not independently verified.
|-
|'''BB(14)''' <br> '''BB(10,3)'''
|'''BBi(26)'''
|style="word-break:break-word"|{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) and BBi(26) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/weakcollatz calr3 on GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| '''Odd multiply perfect'''
| Halts iff an odd [[wikipedia:Multiply perfect number|multiply perfect]] greater than 1 exists.
| '''BB(23,3)'''
| '''BBi(63)'''
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|[https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|Not independently verified.
|-
| '''Odd perfect'''
| Halts iff an odd [[wikipedia:Perfect number|perfect]] exists.
| '''BB(37)'''
| '''BBi(71)'''
|[https://github.com/calr3/tinycryptids/blob/main/oddperfect/machines/oddperfect_37.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/oddperfect calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Sierpiński'''
|Halts iff 78,557 is the smallest [[wikipedia:Sierpiński number|Sierpiński number]].
| '''BB(60)'''
| '''BBi(104)'''
|[https://github.com/calr3/tinycryptids/blob/main/sierpinski/machines/sierpinski_60.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/sierpinski/ calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Riesel'''
|Halts iff 509,203 is the smallest [[wikipedia:Riesel number|Riesel number]].
| '''BB(68)'''
| '''BBi(112)'''
|[https://github.com/loader3229/tinycryptids/blob/main/riesel/machines/riesel_min.txt Machine code]
|[https://github.com/loader3229/tinycryptids/tree/main/riesel loader3229 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]] and DeepSeek
|Includes Lean formalization. Not independently verified.
|-
| rowspan="2"| '''Fermat's Last Theorem'''
| rowspan="2"|Does not halt.
|BB(400)
|
| Not provided.
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994; it is now known that it cannot halt. 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.
|-
| '''BB(90)'''
| '''BBi(160)'''
|[https://github.com/calr3/tinycryptids/blob/main/fermat/machines/fermat_90.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/fermat calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Bigfoot''' - compiled
| Halts iff [[Bigfoot]] halts.
| '''[[BB(7)]]'''
| '''BBi(13)'''
| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot comment]
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states].
|-
| '''Hydra''' - compiled
| Halts iff [[Hydra]] halts.
| '''BB(9)'''
| '''BBi(15)'''
|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}
|[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|}

== Cryptids for other BB-style functions ==

=== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement but not a Π<sub>1</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Source !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{WrapTM|Machine code|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|Found "in the wild".
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting.
|-
|Fermat Primes
|[[Beeping Busy Beaver|BBB(16)]]
|{{WrapTM|Machine code (beep state: P)|1LO1LA_0LI0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RH1RG_1RB1RH_---0LJ_1LK1LJ_1RA0LL_0LM0LL_0RN1RO_1RN1LA_1RO1RP_0LA0RB|non_halt}}
|[https://github.com/calr3/tinycryptids/tree/main/fermatprimesinf calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if there are only finitely many Fermat primes. Not independently verified.
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|[https://github.com/calr3/tinycryptids/tree/main/twinprimes calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|[https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md Mira-Cult GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

=== BitChanger ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|}

=== Boolfuck ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|}

=== CounterScript ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|}

=== Fractran ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|}

=== Lambda Calculus ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|}

=== Pebble Automaton ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|}

=== Post Tag System ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|}

=== Register machine ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(130)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|}

=== Wang Tiles ===

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-26T19:59:45Z

Frogmatic: /* Larger Cryptids */ Clarify status of the BB(215) ZF machine; it was actually posted by Cleo, only the Lean-free proof is by Elliot Glazer.

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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". The best known bounds for each problem (with respect to BB(...), BB(..., n), and [[BBi]](...)) are shown in bold.

{| class="wikitable"
|-
! Name !! Behaviour !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Source !! Date !! Discoverer !! Note
|-
|rowspan="2"| '''[[Logical independence|ZF consistency]]'''
|rowspan="2"| Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|BB(432)
|
|[https://codeberg.org/ajwade/turing_machine_explorer/src/branch/main/zf2.tm Wade's machine]
|[https://codeberg.org/ajwade/turing_machine_explorer ajwade on codeberg.org]
|style="white-space: nowrap; text-align: right;"|Aug 2025
|Wade, based on work by CatIsFluffy and O'Rear
|This [[Logical independence#Axiom_of_Choice|also covers ZFC]].
|-
|'''BB(215)'''
|'''BBi(430)'''
|[https://github.com/VladimirReshetnikov/ProveIt/blob/main/Computability/BusyBeaver/Research/ZFC/k225-formal/lean215/k215_k230labels.tm Machine code]
|[https://github.com/VladimirReshetnikov/ProveIt/tree/main/Computability/BusyBeaver/Research/ZFC/k225-formal/verification/k215 VladimirReshetnikov on GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[wikipedia:Cleo (mathematician)|Vladimir Reshetnikov]]
|Machine-checked in Peano-Arithmetic by [https://github.com/elliotglazer/k215-pa elliotglazer Elliot Glazer].
|-
|'''Peano Arithmetic consistency'''
|Halts iff [[wikipedia:Peano arithmetic|Peano arithmetic]] is inconsistent.
|'''BB(372)'''
|'''BBi(742)'''
|
|[https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py LegionMammal978 GitHub]
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | '''Riemann Hypothesis'''
| rowspan="3" | Halts iff the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|BB(744)
|
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.tm Machine code]
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql metamath-turing-machines GitHub]
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|
|-
|BB(120)
|BBi(238)
|[https://arxiv.org/html/2609.30306v1#A9 Machine code]
|[https://arxiv.org/abs/2609.30306 arxiv]
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|
|-
|'''BB(95)'''
|'''BBi(189)'''
|[https://github.com/google-deepmind/cryptids/blob/main/artifacts/riemann_95/machine.txt Machine code]
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95/ DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"|'''Goldbach'''
| rowspan="2"|Halts iff [[wikipedia:Goldbach's conjecture|Goldbach's conjecture]] is false
|BB(25)
|BBi(49)
|{{WrapTM|Machine code|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|undecided}}
|[https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6 GitHub]
|style="white-space: nowrap; text-align: right;"|May 2016
|anonymous
|Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
|'''BB(21)'''
|'''BBi(41)'''
|{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21 DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"| '''Erdős Ternary Conjecture'''
| rowspan="2"| Halts iff [https://www.erdosproblems.com/406 Erdős Problem 406] is false.
| BB(15) <br> '''BB(5,4)'''
| '''BBi(19)'''
|{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
{{WrapTM|BB(5,4) and BBi(19) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|[https://docs.bbchallenge.org/other/powers_of_two_15_2.txt bbchallenge], [https://docs.bbchallenge.org/other/powers_of_two_5_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
| [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods
| Erdős #406 is the conjecture that "For all n > 8, there is at least one 2 in the base-3 representation of 2^n." [https://arxiv.org/abs/2107.12475 arxiv preprint]
|-
| '''BB(14)'''
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
| [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|Not independently verified.
|-
| '''Erdős Ternary Conjecture''' variant
| Halts iff the following conjecture (a variant of #406) is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1."
| '''BB(4,4)'''
| '''BBi(15)'''
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
| [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
| Not independently verified.
|-
| '''Sloane Ternary Conjecture'''
| Halts if and only if the following [http://neilsloane.com/doc/persistence.html conjecture by N. J. A. Sloane] is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n."
| '''BB(6,4)'''
| '''BBi(21)'''
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
| Not independently verified.
|-
| '''Ternary Conjecture''' variant
| Halts if and only if a n such that 17*2^n has no digit 2 in base-3 exists.
| '''BB(15)'''
| '''BBi(29)'''
|{{WrapTM|Machine code|1RB1LB_1LA1RN_0RE1RD_0RC1RC_0LH1RC_0LG1LH_---1LF_1LF0LL_0LJ1LK_0RD1LI_1LI0LL_1LM1LN_0LI0LI_0RO1LL_1LL0LO|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|
|-
| rowspan="2"| '''Weak Collatz'''
| rowspan="2"|Halts iff the weak [[wikipedia:Collatz conjecture|Collatz conjecture]] is false.
|BB(124) <br> BB(43,4)
|BBi(121)
|
|[https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt bbchallenge], [https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). This is a weakened version of the usual Collatz conjecture, since it does not exclude the possibility of integers that are not part of any cycle at all. Not independently verified.
|-
|'''BB(14)''' <br> '''BB(10,3)'''
|'''BBi(26)'''
|style="word-break:break-word"|{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) and BBi(26) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/weakcollatz calr3 on GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| '''Odd multiply perfect'''
| Halts iff an odd [[wikipedia:Multiply perfect number|multiply perfect]] greater than 1 exists.
| '''BB(23,3)'''
| '''BBi(63)'''
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|[https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|Not independently verified.
|-
| '''Odd perfect'''
| Halts iff an odd [[wikipedia:Perfect number|perfect]] exists.
| '''BB(37)'''
| '''BBi(71)'''
|[https://github.com/calr3/tinycryptids/blob/main/oddperfect/machines/oddperfect_37.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/oddperfect calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Sierpiński'''
|Halts iff 78,557 is the smallest [[wikipedia:Sierpiński number|Sierpiński number]].
| '''BB(60)'''
| '''BBi(104)'''
|[https://github.com/calr3/tinycryptids/blob/main/sierpinski/machines/sierpinski_60.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/sierpinski/ calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Riesel'''
|Halts iff 509,203 is the smallest [[wikipedia:Riesel number|Riesel number]].
| '''BB(68)'''
| '''BBi(112)'''
|[https://github.com/loader3229/tinycryptids/blob/main/riesel/machines/riesel_min.txt Machine code]
|[https://github.com/loader3229/tinycryptids/tree/main/riesel loader3229 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]] and DeepSeek
|Includes Lean formalization. Not independently verified.
|-
| rowspan="2"| '''Fermat's Last Theorem'''
| rowspan="2"|Does not halt.
|BB(400)
|
| Not provided.
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994; it is now known that it cannot halt. 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.
|-
| '''BB(90)'''
| '''BBi(160)'''
|[https://github.com/calr3/tinycryptids/blob/main/fermat/machines/fermat_90.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/fermat calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Bigfoot''' - compiled
| Halts iff [[Bigfoot]] halts.
| '''[[BB(7)]]'''
| '''BBi(13)'''
| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot comment]
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states].
|-
| '''Hydra''' - compiled
| Halts iff [[Hydra]] halts.
| '''BB(9)'''
| '''BBi(15)'''
|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}
|[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|-
| '''Antihydra-like'''
|
| '''[[BB(7)]]'''
| '''BBi(13)'''
|{{WrapTM|Machine code|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|2n -> 3n, 2n+1 -> 3n+5, start at 2, halts only when the number of odd numbers is more than twice the number of even numbers.
|}

== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Source !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{WrapTM|Machine code|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|Found "in the wild".
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting.
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|[https://github.com/calr3/tinycryptids/tree/main/twinprimes calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|[https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md Mira-Cult GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-26T19:20:42Z

Frogmatic: /* Larger Cryptids */ Merge ZF and ZFC rows; these were incorrectly represented as two separate problems, but logical independence explains why they're equivalent from a halting perspective.

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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". The best known bounds for each problem (with respect to BB(...), BB(..., n), and [[BBi]](...)) are shown in bold.

{| class="wikitable"
|-
! Name !! Behaviour !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Source !! Date !! Discoverer !! Note
|-
|rowspan="2"| '''[[Logical independence|ZF consistency]]'''
|rowspan="2"| Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|BB(432)
|
|[https://codeberg.org/ajwade/turing_machine_explorer/src/branch/main/zf2.tm Wade's machine]
|[https://codeberg.org/ajwade/turing_machine_explorer ajwade on codeberg.org]
|style="white-space: nowrap; text-align: right;"|Aug 2025
|Wade, based on work by CatIsFluffy and O'Rear
|This [[Logical independence#Axiom_of_Choice|also covers ZFC]].
|-
|'''BB(215)'''
|
|
|[https://github.com/elliotglazer/k215-pa elliotglazer GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Elliot Glazer
|Machine-checked in Peano-Arithmetic. Not independently verified.
|-
|'''Peano Arithmetic consistency'''
|Halts iff [[wikipedia:Peano arithmetic|Peano arithmetic]] is inconsistent.
|'''BB(372)'''
|'''BBi(742)'''
|
|[https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py LegionMammal978 GitHub]
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | '''Riemann Hypothesis'''
| rowspan="3" | Halts iff the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|BB(744)
|
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.tm Machine code]
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql metamath-turing-machines GitHub]
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|
|-
|BB(120)
|BBi(238)
|[https://arxiv.org/html/2609.30306v1#A9 Machine code]
|[https://arxiv.org/abs/2609.30306 arxiv]
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|
|-
|'''BB(95)'''
|'''BBi(189)'''
|[https://github.com/google-deepmind/cryptids/blob/main/artifacts/riemann_95/machine.txt Machine code]
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95/ DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"|'''Goldbach'''
| rowspan="2"|Halts iff [[wikipedia:Goldbach's conjecture|Goldbach's conjecture]] is false
|BB(25)
|BBi(49)
|{{WrapTM|Machine code|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|undecided}}
|[https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6 GitHub]
|style="white-space: nowrap; text-align: right;"|May 2016
|anonymous
|Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
|'''BB(21)'''
|'''BBi(41)'''
|{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21 DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"| '''Erdős Ternary Conjecture'''
| rowspan="2"| Halts iff [https://www.erdosproblems.com/406 Erdős Problem 406] is false.
| BB(15) <br> '''BB(5,4)'''
| '''BBi(19)'''
|{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
{{WrapTM|BB(5,4) and BBi(19) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|[https://docs.bbchallenge.org/other/powers_of_two_15_2.txt bbchallenge], [https://docs.bbchallenge.org/other/powers_of_two_5_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
| [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods
| Erdős #406 is the conjecture that "For all n > 8, there is at least one 2 in the base-3 representation of 2^n." [https://arxiv.org/abs/2107.12475 arxiv preprint]
|-
| '''BB(14)'''
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
| [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|Not independently verified.
|-
| '''Erdős Ternary Conjecture''' variant
| Halts iff the following conjecture (a variant of #406) is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1."
| '''BB(4,4)'''
| '''BBi(15)'''
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
| [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
| Not independently verified.
|-
| '''Sloane Ternary Conjecture'''
| Halts if and only if the following [http://neilsloane.com/doc/persistence.html conjecture by N. J. A. Sloane] is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n."
| '''BB(6,4)'''
| '''BBi(21)'''
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
| Not independently verified.
|-
| '''Ternary Conjecture''' variant
| Halts if and only if a n such that 17*2^n has no digit 2 in base-3 exists.
| '''BB(15)'''
| '''BBi(29)'''
|{{WrapTM|Machine code|1RB1LB_1LA1RN_0RE1RD_0RC1RC_0LH1RC_0LG1LH_---1LF_1LF0LL_0LJ1LK_0RD1LI_1LI0LL_1LM1LN_0LI0LI_0RO1LL_1LL0LO|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|
|-
| rowspan="2"| '''Weak Collatz'''
| rowspan="2"|Halts iff the weak [[wikipedia:Collatz conjecture|Collatz conjecture]] is false.
|BB(124) <br> BB(43,4)
|BBi(121)
|
|[https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt bbchallenge], [https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). This is a weakened version of the usual Collatz conjecture, since it does not exclude the possibility of integers that are not part of any cycle at all. Not independently verified.
|-
|'''BB(14)''' <br> '''BB(10,3)'''
|'''BBi(26)'''
|style="word-break:break-word"|{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) and BBi(26) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/weakcollatz calr3 on GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| '''Odd multiply perfect'''
| Halts iff an odd [[wikipedia:Multiply perfect number|multiply perfect]] greater than 1 exists.
| '''BB(23,3)'''
| '''BBi(63)'''
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|[https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|Not independently verified.
|-
| '''Odd perfect'''
| Halts iff an odd [[wikipedia:Perfect number|perfect]] exists.
| '''BB(37)'''
| '''BBi(71)'''
|[https://github.com/calr3/tinycryptids/blob/main/oddperfect/machines/oddperfect_37.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/oddperfect calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Sierpiński'''
|Halts iff 78,557 is the smallest [[wikipedia:Sierpiński number|Sierpiński number]].
| '''BB(60)'''
| '''BBi(104)'''
|[https://github.com/calr3/tinycryptids/blob/main/sierpinski/machines/sierpinski_60.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/sierpinski/ calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Riesel'''
|Halts iff 509,203 is the smallest [[wikipedia:Riesel number|Riesel number]].
| '''BB(68)'''
| '''BBi(112)'''
|[https://github.com/loader3229/tinycryptids/blob/main/riesel/machines/riesel_min.txt Machine code]
|[https://github.com/loader3229/tinycryptids/tree/main/riesel loader3229 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]] and DeepSeek
|Includes Lean formalization. Not independently verified.
|-
| rowspan="2"| '''Fermat's Last Theorem'''
| rowspan="2"|Does not halt.
|BB(400)
|
| Not provided.
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994; it is now known that it cannot halt. 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.
|-
| '''BB(90)'''
| '''BBi(160)'''
|[https://github.com/calr3/tinycryptids/blob/main/fermat/machines/fermat_90.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/fermat calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Bigfoot''' - compiled
| Halts iff [[Bigfoot]] halts.
| '''[[BB(7)]]'''
| '''BBi(13)'''
| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot comment]
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states].
|-
| '''Hydra''' - compiled
| Halts iff [[Hydra]] halts.
| '''BB(9)'''
| '''BBi(15)'''
|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}
|[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|-
| '''Antihydra-like'''
|
| '''[[BB(7)]]'''
| '''BBi(13)'''
|{{WrapTM|Machine code|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|2n -> 3n, 2n+1 -> 3n+5, start at 2, halts only when the number of odd numbers is more than twice the number of even numbers.
|}

== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Source !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{WrapTM|Machine code|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|Found "in the wild".
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting.
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|[https://github.com/calr3/tinycryptids/tree/main/twinprimes calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|[https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md Mira-Cult GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T15:56:57Z

Frogmatic: /* Larger Cryptids */ Add more context links for problems

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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". The best known bounds for each problem (with respect to BB(...), BB(..., n), and [[BBi]](...)) are shown in bold.

{| class="wikitable"
|-
! Name !! Behaviour !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Source !! Date !! Discoverer !! Note
|-
|'''[[Logical independence|ZF consistency]]'''
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|'''BB(432)'''
|
|[https://codeberg.org/ajwade/turing_machine_explorer/src/branch/main/zf2.tm Wade's machine]
|[https://codeberg.org/ajwade/turing_machine_explorer ajwade on codeberg.org]
|style="white-space: nowrap; text-align: right;"|2025
|Wade, based on work by CatIsFluffy and O'Rear
|
|-
|'''ZFC consistency'''
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|'''BB(215)'''
|
|
|[https://github.com/elliotglazer/k215-pa elliotglazer GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Elliot Glazer
|Machine-checked in Peano-Arithmetic. Not independently verified.
|-
|'''Peano Arithmetic consistency'''
|Halts iff [[wikipedia:Peano arithmetic|Peano arithmetic]] is inconsistent.
|'''BB(372)'''
|'''BBi(742)'''
|
|[https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py LegionMammal978 GitHub]
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | '''Riemann Hypothesis'''
| rowspan="3" | Halts iff the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|BB(744)
|
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.tm Machine code]
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql metamath-turing-machines GitHub]
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|
|-
|BB(120)
|BBi(238)
|[https://arxiv.org/html/2609.30306v1#A9 Machine code]
|[https://arxiv.org/abs/2609.30306 arxiv]
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|
|-
|'''BB(95)'''
|'''BBi(189)'''
|[https://github.com/google-deepmind/cryptids/blob/main/artifacts/riemann_95/machine.txt Machine code]
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95/ DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"|'''Goldbach'''
| rowspan="2"|Halts iff [[wikipedia:Goldbach's conjecture|Goldbach's conjecture]] is false
|BB(25)
|BBi(49)
|{{WrapTM|Machine code|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|undecided}}
|[https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6 GitHub]
|style="white-space: nowrap; text-align: right;"|May 2016
|anonymous
|Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
|'''BB(21)'''
|'''BBi(41)'''
|{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21 DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"| '''Erdős Ternary Conjecture'''
| rowspan="2"| Halts iff [https://www.erdosproblems.com/406 Erdős Problem 406] is false.
| BB(15) <br> '''BB(5,4)'''
| '''BBi(19)'''
|{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
{{WrapTM|BB(5,4) and BBi(19) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|[https://docs.bbchallenge.org/other/powers_of_two_15_2.txt bbchallenge], [https://docs.bbchallenge.org/other/powers_of_two_5_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
| [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods
| Erdős #406 is the conjecture that "For all n > 8, there is at least one 2 in the base-3 representation of 2^n." [https://arxiv.org/abs/2107.12475 arxiv preprint]
|-
| '''BB(14)'''
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
| [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|Not independently verified.
|-
| '''Erdős Ternary Conjecture''' variant
| Halts iff the following conjecture (a variant of #406) is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1."
| '''BB(4,4)'''
| '''BBi(15)'''
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
| [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
| Not independently verified.
|-
| '''Sloane Ternary Conjecture'''
| Halts if and only if the following [http://neilsloane.com/doc/persistence.html conjecture by N. J. A. Sloane] is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n."
| '''BB(6,4)'''
| '''BBi(21)'''
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
| Not independently verified.
|-
| rowspan="2"| '''Weak Collatz'''
| rowspan="2"|Halts iff the weak [[wikipedia:Collatz conjecture|Collatz conjecture]] is false.
|BB(124) <br> BB(43,4)
|BBi(121)
|
|[https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt bbchallenge], [https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). This is a weakened version of the usual Collatz conjecture, since it does not exclude the possibility of integers that are not part of any cycle at all. Not independently verified.
|-
|'''BB(14)''' <br> '''BB(10,3)'''
|'''BBi(26)'''
|style="word-break:break-word"|{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) and BBi(26) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/weakcollatz calr3 on GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| '''Odd multiply perfect'''
| Halts iff an odd [[wikipedia:Multiply perfect number|multiply perfect]] greater than 1 exists.
| '''BB(23,3)'''
| '''BBi(63)'''
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|[https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|Not independently verified.
|-
| '''Odd perfect'''
| Halts iff an odd [[wikipedia:Perfect number|perfect]] exists.
| '''BB(37)'''
| '''BBi(71)'''
|[https://github.com/calr3/tinycryptids/blob/main/oddperfect/machines/oddperfect_37.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/oddperfect calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Sierpiński'''
|Halts iff 78,557 is the smallest [[wikipedia:Sierpiński number|Sierpiński number]].
| '''BB(60)'''
| '''BBi(104)'''
|[https://github.com/calr3/tinycryptids/blob/main/sierpinski/machines/sierpinski_60.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/sierpinski/ calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Riesel'''
|Halts iff 509,203 is the smallest [[wikipedia:Riesel number|Riesel number]].
| '''BB(68)'''
| '''BBi(112)'''
|[https://github.com/loader3229/tinycryptids/blob/main/riesel/machines/riesel_min.txt Machine code]
|[https://github.com/loader3229/tinycryptids/tree/main/riesel loader3229 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|Includes Lean formalization. Not independently verified.
|-
| rowspan="2"| '''Fermat's Last Theorem'''
| rowspan="2"|Does not halt.
|BB(400)
|
| Not provided.
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994; it is now known that it cannot halt. 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.
|-
| '''BB(90)'''
| '''BBi(160)'''
|[https://github.com/calr3/tinycryptids/blob/main/fermat/machines/fermat_90.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/fermat calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Bigfoot''' - compiled
| Halts iff [[Bigfoot]] halts.
| '''[[BB(7)]]'''
| '''BBi(13)'''
| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot comment]
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states].
|-
| '''Hydra''' - compiled
| Halts iff [[Hydra]] halts.
| '''BB(9)'''
| '''BBi(15)'''
|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}
|[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|-
| '''Antihydra-like'''
|
| '''[[BB(7)]]'''
| '''BBi(13)'''
|{{WrapTM|Machine code|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|2n -> 3n, 2n+1 -> 3n+5, start at 2, halts only when the number of odd numbers is more than twice the number of even numbers.
|}

== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Source !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{WrapTM|Machine code|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|Found "in the wild".
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting.
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|[https://github.com/calr3/tinycryptids/tree/main/twinprimes calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|[https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md Mira-Cult GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T15:52:38Z

Frogmatic: /* Beeping Busy Beaver */ Split source info out from the machine itself, as for the table above.

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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". The best known bounds for each problem (with respect to BB(...), BB(..., n), and [[BBi]](...)) are shown in bold.

{| class="wikitable"
|-
! Name !! Behaviour !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Source !! Date !! Discoverer !! Note
|-
|'''[[Logical independence|ZF consistency]]'''
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|'''BB(432)'''
|
|[https://codeberg.org/ajwade/turing_machine_explorer/src/branch/main/zf2.tm Wade's machine]
|[https://codeberg.org/ajwade/turing_machine_explorer ajwade on codeberg.org]
|style="white-space: nowrap; text-align: right;"|2025
|Wade, based on work by CatIsFluffy and O'Rear
|
|-
|'''ZFC consistency'''
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|'''BB(215)'''
|
|
|[https://github.com/elliotglazer/k215-pa elliotglazer GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Elliot Glazer
|Machine-checked in Peano-Arithmetic. Not independently verified.
|-
|'''Peano Arithmetic consistency'''
|Halts iff Peano Arithmetic is inconsistent
|'''BB(372)'''
|'''BBi(742)'''
|
|[https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py LegionMammal978 GitHub]
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | '''Riemann Hypothesis'''
| rowspan="3" | Halts iff the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|BB(744)
|
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.tm Machine code]
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql metamath-turing-machines GitHub]
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|
|-
|BB(120)
|BBi(238)
|[https://arxiv.org/html/2609.30306v1#A9 Machine code]
|[https://arxiv.org/abs/2609.30306 arxiv]
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|
|-
|'''BB(95)'''
|'''BBi(189)'''
|[https://github.com/google-deepmind/cryptids/blob/main/artifacts/riemann_95/machine.txt Machine code]
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95/ DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"|'''Goldbach'''
| rowspan="2"|Halts iff [[wikipedia:Goldbach's conjecture|Goldbach's conjecture]] is false
|BB(25)
|BBi(49)
|{{WrapTM|Machine code|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|undecided}}
|[https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6 GitHub]
|style="white-space: nowrap; text-align: right;"|May 2016
|anonymous
|Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
|'''BB(21)'''
|'''BBi(41)'''
|{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21 DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"| '''Erdős Ternary Conjecture'''
| rowspan="2"| Halts iff [https://www.erdosproblems.com/406 Erdős Problem 406] is false.
| BB(15) <br> '''BB(5,4)'''
| '''BBi(19)'''
|{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
{{WrapTM|BB(5,4) and BBi(19) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|[https://docs.bbchallenge.org/other/powers_of_two_15_2.txt bbchallenge], [https://docs.bbchallenge.org/other/powers_of_two_5_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
| [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods
| Erdős #406 is the conjecture that "For all n > 8, there is at least one 2 in the base-3 representation of 2^n." [https://arxiv.org/abs/2107.12475 arxiv preprint]
|-
| '''BB(14)'''
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
| [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|Not independently verified.
|-
| '''Erdős Ternary Conjecture''' variant
| Halts iff the following conjecture (a variant of #406) is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1."
| '''BB(4,4)'''
| '''BBi(15)'''
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
| [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
| Not independently verified.
|-
| '''Sloane Ternary Conjecture'''
| Halts if and only if the following [http://neilsloane.com/doc/persistence.html conjecture by N. J. A. Sloane] is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n."
| '''BB(6,4)'''
| '''BBi(21)'''
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
| Not independently verified.
|-
| rowspan="2"| '''Weak Collatz'''
| rowspan="2"|Halts iff the weak Collatz conjecture is false.
|BB(124) <br> BB(43,4)
|BBi(121)
|
|[https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt bbchallenge], [https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). Not independently verified, and probably easy to further optimise.
|-
|'''BB(14)''' <br> '''BB(10,3)'''
|'''BBi(26)'''
|style="word-break:break-word"|{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) and BBi(26) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/weakcollatz calr3 on GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| '''Odd multiply perfect'''
| Halts iff an odd [[wikipedia:Multiply perfect number|multiply perfect]] greater than 1 exists.
| '''BB(23,3)'''
| '''BBi(63)'''
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|[https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|Not independently verified.
|-
| '''Odd perfect'''
| Halts iff an odd [[wikipedia:Perfect number|perfect]] exists.
| '''BB(37)'''
| '''BBi(71)'''
|[https://github.com/calr3/tinycryptids/blob/main/oddperfect/machines/oddperfect_37.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/oddperfect calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Sierpiński'''
|Halts iff 78,557 is the smallest [[wikipedia:Sierpiński number|Sierpiński number]].
| '''BB(60)'''
| '''BBi(104)'''
|[https://github.com/calr3/tinycryptids/blob/main/sierpinski/machines/sierpinski_60.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/sierpinski/ calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Riesel'''
|Halts iff 509,203 is the smallest [[wikipedia:Riesel number|Riesel number]].
| '''BB(68)'''
| '''BBi(112)'''
|[https://github.com/loader3229/tinycryptids/blob/main/riesel/machines/riesel_min.txt Machine code]
|[https://github.com/loader3229/tinycryptids/tree/main/riesel loader3229 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|Includes Lean formalization. Not independently verified.
|-
| rowspan="2"| '''Fermat's Last Theorem'''
| rowspan="2"|Does not halt.
|BB(400)
|
| Not provided.
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994; it is now known that it cannot halt. 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.
|-
| '''BB(90)'''
| '''BBi(160)'''
|[https://github.com/calr3/tinycryptids/blob/main/fermat/machines/fermat_90.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/fermat calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Bigfoot''' - compiled
| Halts iff [[Bigfoot]] halts.
| '''[[BB(7)]]'''
| '''BBi(13)'''
| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot comment]
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states].
|-
| '''Hydra''' - compiled
| Halts iff [[Hydra]] halts.
| '''BB(9)'''
| '''BBi(15)'''
|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}
|[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|-
| '''Antihydra-like'''
|
| '''[[BB(7)]]'''
| '''BBi(13)'''
|{{WrapTM|Machine code|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|2n -> 3n, 2n+1 -> 3n+5, start at 2, halts only when the number of odd numbers is more than twice the number of even numbers.
|}

== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Source !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{WrapTM|Machine code|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|Found "in the wild".
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting.
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|[https://github.com/calr3/tinycryptids/tree/main/twinprimes calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|[https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md Mira-Cult GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T15:50:36Z

Frogmatic: /* Larger Cryptids */ Split out two concepts: the machine itself (where available) and the full external source. In some cases I couldn't actually find the real machine, but I think it's useful to highlight that for further digging.

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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". The best known bounds for each problem (with respect to BB(...), BB(..., n), and [[BBi]](...)) are shown in bold.

{| class="wikitable"
|-
! Name !! Behaviour !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Source !! Date !! Discoverer !! Note
|-
|'''[[Logical independence|ZF consistency]]'''
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|'''BB(432)'''
|
|[https://codeberg.org/ajwade/turing_machine_explorer/src/branch/main/zf2.tm Wade's machine]
|[https://codeberg.org/ajwade/turing_machine_explorer ajwade on codeberg.org]
|style="white-space: nowrap; text-align: right;"|2025
|Wade, based on work by CatIsFluffy and O'Rear
|
|-
|'''ZFC consistency'''
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|'''BB(215)'''
|
|
|[https://github.com/elliotglazer/k215-pa elliotglazer GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Elliot Glazer
|Machine-checked in Peano-Arithmetic. Not independently verified.
|-
|'''Peano Arithmetic consistency'''
|Halts iff Peano Arithmetic is inconsistent
|'''BB(372)'''
|'''BBi(742)'''
|
|[https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py LegionMammal978 GitHub]
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | '''Riemann Hypothesis'''
| rowspan="3" | Halts iff the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|BB(744)
|
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.tm Machine code]
|[https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql metamath-turing-machines GitHub]
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|
|-
|BB(120)
|BBi(238)
|[https://arxiv.org/html/2609.30306v1#A9 Machine code]
|[https://arxiv.org/abs/2609.30306 arxiv]
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|
|-
|'''BB(95)'''
|'''BBi(189)'''
|[https://github.com/google-deepmind/cryptids/blob/main/artifacts/riemann_95/machine.txt Machine code]
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95/ DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"|'''Goldbach'''
| rowspan="2"|Halts iff [[wikipedia:Goldbach's conjecture|Goldbach's conjecture]] is false
|BB(25)
|BBi(49)
|{{WrapTM|Machine code|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|undecided}}
|[https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6 GitHub]
|style="white-space: nowrap; text-align: right;"|May 2016
|anonymous
|Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
|'''BB(21)'''
|'''BBi(41)'''
|{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|[https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21 DeepMind GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"| '''Erdős Ternary Conjecture'''
| rowspan="2"| Halts iff [https://www.erdosproblems.com/406 Erdős Problem 406] is false.
| BB(15) <br> '''BB(5,4)'''
| '''BBi(19)'''
|{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
{{WrapTM|BB(5,4) and BBi(19) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|[https://docs.bbchallenge.org/other/powers_of_two_15_2.txt bbchallenge], [https://docs.bbchallenge.org/other/powers_of_two_5_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
| [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods
| Erdős #406 is the conjecture that "For all n > 8, there is at least one 2 in the base-3 representation of 2^n." [https://arxiv.org/abs/2107.12475 arxiv preprint]
|-
| '''BB(14)'''
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
| [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|Not independently verified.
|-
| '''Erdős Ternary Conjecture''' variant
| Halts iff the following conjecture (a variant of #406) is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1."
| '''BB(4,4)'''
| '''BBi(15)'''
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
| [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
| Not independently verified.
|-
| '''Sloane Ternary Conjecture'''
| Halts if and only if the following [http://neilsloane.com/doc/persistence.html conjecture by N. J. A. Sloane] is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n."
| '''BB(6,4)'''
| '''BBi(21)'''
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
| Not independently verified.
|-
| rowspan="2"| '''Weak Collatz'''
| rowspan="2"|Halts iff the weak Collatz conjecture is false.
|BB(124) <br> BB(43,4)
|BBi(121)
|
|[https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt bbchallenge], [https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt bbchallenge]
|style="white-space: nowrap; text-align: right;"|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). Not independently verified, and probably easy to further optimise.
|-
|'''BB(14)''' <br> '''BB(10,3)'''
|'''BBi(26)'''
|style="word-break:break-word"|{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) and BBi(26) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/weakcollatz calr3 on GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| '''Odd multiply perfect'''
| Halts iff an odd [[wikipedia:Multiply perfect number|multiply perfect]] greater than 1 exists.
| '''BB(23,3)'''
| '''BBi(63)'''
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|[https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|Not independently verified.
|-
| '''Odd perfect'''
| Halts iff an odd [[wikipedia:Perfect number|perfect]] exists.
| '''BB(37)'''
| '''BBi(71)'''
|[https://github.com/calr3/tinycryptids/blob/main/oddperfect/machines/oddperfect_37.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/oddperfect calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Sierpiński'''
|Halts iff 78,557 is the smallest [[wikipedia:Sierpiński number|Sierpiński number]].
| '''BB(60)'''
| '''BBi(104)'''
|[https://github.com/calr3/tinycryptids/blob/main/sierpinski/machines/sierpinski_60.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/sierpinski/ calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Riesel'''
|Halts iff 509,203 is the smallest [[wikipedia:Riesel number|Riesel number]].
| '''BB(68)'''
| '''BBi(112)'''
|[https://github.com/loader3229/tinycryptids/blob/main/riesel/machines/riesel_min.txt Machine code]
|[https://github.com/loader3229/tinycryptids/tree/main/riesel loader3229 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|Includes Lean formalization. Not independently verified.
|-
| rowspan="2"| '''Fermat's Last Theorem'''
| rowspan="2"|Does not halt.
|BB(400)
|
| Not provided.
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994; it is now known that it cannot halt. 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.
|-
| '''BB(90)'''
| '''BBi(160)'''
|[https://github.com/calr3/tinycryptids/blob/main/fermat/machines/fermat_90.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/fermat calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Bigfoot''' - compiled
| Halts iff [[Bigfoot]] halts.
| '''[[BB(7)]]'''
| '''BBi(13)'''
| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot comment]
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states].
|-
| '''Hydra''' - compiled
| Halts iff [[Hydra]] halts.
| '''BB(9)'''
| '''BBi(15)'''
|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}
|[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|-
| '''Antihydra-like'''
|
| '''[[BB(7)]]'''
| '''BBi(13)'''
|{{WrapTM|Machine code|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|2n -> 3n, 2n+1 -> 3n+5, start at 2, halts only when the number of odd numbers is more than twice the number of even numbers.
|}

== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|https://github.com/calr3/tinycryptids/tree/main/twinprimes
{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T15:17:16Z

Frogmatic: /* Larger Cryptids */ Highlight best bounds in bold

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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". The best known bounds for each problem (with respect to BB(...), BB(..., n), and [[BBi]](...)) are shown in bold.

{| class="wikitable"
|-
! Name !! Behaviour !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
|-
|'''[[Logical independence|ZF consistency]]'''
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|'''BB(432)'''
|
|style="word-break:break-word"|[https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e Wade's machine]
|style="white-space: nowrap; text-align: right;"|2025
|Wade, based on work by CatIsFluffy and O'Rear
|
|-
|'''ZFC consistency'''
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|'''BB(215)'''
|
|style="word-break:break-word"|https://github.com/elliotglazer/k215-pa
|style="white-space: nowrap; text-align: right;"|Oct 2026
|elliotglazer
|Machine-checked in Peano-Arithmetic. Not independently verified.
|-
|'''Peano Arithmetic consistency'''
|Halts iff Peano Arithmetic is inconsistent
|'''BB(372)'''
|'''BBi(742)'''
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | '''Riemann Hypothesis'''
| rowspan="3" | Halts iff the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|BB(744)
|
|style="word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|
|-
|BB(120)
|BBi(238)
|style="word-break:break-word"|https://arxiv.org/abs/2609.30306
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|
|-
|'''BB(95)'''
|'''BBi(189)'''
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"|'''Goldbach'''
| rowspan="2"|Halts iff [[wikipedia:Goldbach's conjecture|Goldbach's conjecture]] is false
|BB(25)
|BBi(49)
|style="word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
{{WrapTM|Machine code|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|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2016
|anonymous
|Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
|'''BB(21)'''
|'''BBi(41)'''
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"| '''Erdős Ternary Conjecture'''
| rowspan="2"| Halts iff [https://www.erdosproblems.com/406 Erdős Problem 406] is false.
| BB(15) <br> '''BB(5,4)'''
| '''BBi(19)'''
|style="word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
{{WrapTM|BB(5,4) and BBi(19) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2021
| [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods
| Erdős #406 is the conjecture that "For all n > 8, there is at least one 2 in the base-3 representation of 2^n." [https://arxiv.org/abs/2107.12475 arxiv preprint]
|-
| '''BB(14)'''
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| '''Erdős Ternary Conjecture''' variant
| Halts iff the following conjecture (a variant of #406) is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1."
| '''BB(4,4)'''
| '''BBi(15)'''
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
| Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| '''Sloane Ternary Conjecture'''
| Halts if and only if the following [http://neilsloane.com/doc/persistence.html conjecture by N. J. A. Sloane] is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n."
| '''BB(6,4)'''
| '''BBi(21)'''
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
| Not independently verified.
|-
| rowspan="2"| '''Weak Collatz'''
| rowspan="2"|Halts iff the weak Collatz conjecture is false.
|BB(124) <br> '''BB(43,4)'''
|BBi(121)
|style="word-break:break-word"|https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt
https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt
|style="white-space: nowrap; text-align: right;"|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). Not independently verified, and probably easy to further optimise.
|-
|'''BB(14)''' <br> '''BB(10,3)'''
|'''BBi(26)'''
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz
{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) and BBi(26) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| '''Odd multiply perfect'''
| Halts iff an odd [[wikipedia:Multiply perfect number|multiply perfect]] greater than 1 exists.
| '''BB(23,3)'''
| '''BBi(63)'''
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
| '''Odd perfect'''
| Halts iff an odd [[wikipedia:Perfect number|perfect]] exists.
| '''BB(37)'''
| '''BBi(71)'''
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Sierpiński'''
|Halts iff 78,557 is the smallest [[wikipedia:Sierpiński number|Sierpiński number]].
| '''BB(60)'''
| '''BBi(104)'''
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Riesel'''
|Halts iff 509,203 is the smallest [[wikipedia:Riesel number|Riesel number]].
| '''BB(68)'''
| '''BBi(112)'''
|style="word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|Includes Lean formalization. Not independently verified.
|-
| rowspan="2"| '''Fermat's Last Theorem'''
| rowspan="2"|Does not halt.
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994; it is now known that it cannot halt. 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.
|-
| '''BB(90)'''
| '''BBi(160)'''
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| '''Bigfoot''' - compiled
| Halts iff [[Bigfoot]] halts.
| '''[[BB(7)]]'''
| '''BBi(13)'''
| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| '''Hydra''' - compiled
| Halts iff [[Hydra]] halts.
| '''BB(9)'''
| '''BBi(15)'''
|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}
[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|-
| '''Antihydra-like'''
|
| '''[[BB(7)]]'''
| '''BBi(13)'''
|{{WrapTM|Machine code|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|2n -> 3n, 2n+1 -> 3n+5, start at 2, halts only when the number of odd numbers is more than twice the number of even numbers.
|}

== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|https://github.com/calr3/tinycryptids/tree/main/twinprimes
{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T15:06:33Z

Frogmatic: /* Larger Cryptids */ Tidy up what's in the "Machine" column versus Notes, so this column is now a reasonable width.

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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 !! Behaviour !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF consistency]]
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|BB(432)
|
|style="word-break:break-word"|[https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e Wade's machine]
|style="white-space: nowrap; text-align: right;"|2025
|Wade, based on work by CatIsFluffy and O'Rear
|
|-
|ZFC consistency
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|BB(215)
|
|style="word-break:break-word"|https://github.com/elliotglazer/k215-pa
|style="white-space: nowrap; text-align: right;"|Oct 2026
|elliotglazer
|Machine-checked in Peano-Arithmetic. Not independently verified.
|-
|Peano Arithmetic consistency
|Halts iff Peano Arithmetic is inconsistent
|BB(372)
|BBi(742)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | Riemann Hypothesis
| rowspan="3" | Halts iff the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|BB(744)
|
|style="word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|
|-
|BB(120)
|BBi(238)
|style="word-break:break-word"|https://arxiv.org/abs/2609.30306
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|
|-
|BB(95)
|BBi(189)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"|Goldbach
| rowspan="2"|Halts iff [[wikipedia:Goldbach's conjecture|Goldbach's conjecture]] is false
|BB(25)
|BBi(49)
|style="word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
{{WrapTM|Machine code|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|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2016
|anonymous
|Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
|BB(21)
|BBi(41)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"| Erdős Ternary Conjecture
| rowspan="2"| Halts iff [https://www.erdosproblems.com/406 Erdős Problem 406] is false.
| BB(15) <br> BB(5,4)
| BBi(19)
|style="word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
{{WrapTM|BB(5,4) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2021
| [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods
| Erdős #406 is the conjecture that "For all n > 8, there is at least one 2 in the base-3 representation of 2^n." [https://arxiv.org/abs/2107.12475 arxiv preprint]
|-
| BB(14)
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| Erdős Ternary Conjecture Variant
| Halts iff the following conjecture (a variant of #406) is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1."
| BB(4,4)
| BBi(15)
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
| Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| Sloane Ternary Conjecture
| Halts if and only if the following [http://neilsloane.com/doc/persistence.html conjecture by N. J. A. Sloane] is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n."
| BB(6,4)
| BBi(21)
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
| Not independently verified.
|-
| rowspan="2"|Weak Collatz
| rowspan="2"|Halts iff the weak Collatz conjecture is false.
|BB(124) <br> BB(43,4)
|BBi(121)
|style="word-break:break-word"|https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt
https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt
|style="white-space: nowrap; text-align: right;"|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). Not independently verified, and probably easy to further optimise.
|-
|BB(14) <br> BB(10,3)
|BBi(26)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz
{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| Odd multiply perfect
| Halts iff an odd [[wikipedia:Multiply perfect number|multiply perfect]] greater than 1 exists.
|BB(23,3)
|BBi(63)
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
| Odd perfect
| Halts iff an odd [[wikipedia:Perfect number|perfect]] exists.
|BB(37)
|BBi(71)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
|Sierpiński
|Halts iff 78,557 is the smallest [[wikipedia:Sierpiński number|Sierpiński number]].
|BB(60)
|BBi(104)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
|Riesel
|Halts iff 509,203 is the smallest [[wikipedia:Riesel number|Riesel number]].
|BB(68)
|BBi(112)
|style="word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|Includes Lean formalization. Not independently verified.
|-
| rowspan="2"|Fermat's Last Theorem
| rowspan="2"|Does not halt.
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994; it is now known that it cannot halt. 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.
|-
|BB(90)
|BBi(160)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Includes Lean formalization. Not independently verified.
|-
| Bigfoot - compiled
| Halts iff [[Bigfoot]] halts.
| [[BB(7)]]
| BBi(13)
| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| Hydra - compiled
| Halts iff [[Hydra]] halts.
|BB(9)
|BBi(15)
|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}
[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|-
| Antihydra-like
|
|[[BB(7)]]
|BBi(13)
|{{WrapTM|Machine code|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|2n -> 3n, 2n+1 -> 3n+5, start at 2, halts only when the number of odd numbers is more than twice the number of even numbers.
|}

== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|https://github.com/calr3/tinycryptids/tree/main/twinprimes
{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T14:59:24Z

Frogmatic: /* Larger Cryptids */ Move the behaviour description into a dedicated column. This removes the need to keep restating it for machines that address the same problem, and helps to separate the problem statement from the solutions.

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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 !! Behaviour !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF consistency]]
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|BB(432)
|
|style="word-break:break-word"|[https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e Wade's machine]
|style="white-space: nowrap; text-align: right;"|2025
|Wade, based on work by CatIsFluffy and O'Rear
|
|-
|ZFC consistency
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|BB(215)
|
|style="word-break:break-word"|https://github.com/elliotglazer/k215-pa
|style="white-space: nowrap; text-align: right;"|Oct 2026
|elliotglazer
|Machine-checked in Peano-Arithmetic. Not independently verified.
|-
|Peano Arithmetic consistency
|Halts iff Peano Arithmetic is inconsistent
|BB(372)
|BBi(742)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | Riemann Hypothesis
| rowspan="3" | Halts iff the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|BB(744)
|
|style="word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|
|-
|BB(120)
|BBi(238)
|style="word-break:break-word"|https://arxiv.org/abs/2609.30306
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|
|-
|BB(95)
|BBi(189)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"|Goldbach
| rowspan="2"|Halts iff [[wikipedia:Goldbach's conjecture|Goldbach's conjecture]] is false
|BB(25)
|BBi(49)
|style="word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
{{WrapTM|Machine code|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|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2016
|anonymous
|Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
|BB(21)
|BBi(41)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|Not independently verified.
|-
| rowspan="2"| Erdős Ternary Conjecture
| rowspan="2"| Halts iff [https://www.erdosproblems.com/406 Erdős Problem 406] is false.
| BB(15) <br> BB(5,4)
| BBi(19)
|style="word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
{{WrapTM|BB(5,4) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2021
| [[User:Cosmo|Tristan Stérin]] (<code>@cosmo</code>) and Damien Woods
| Erdős #406 is the conjecture that "For all n > 8, there is at least one 2 in the base-3 representation of 2^n." [https://arxiv.org/abs/2107.12475 arxiv preprint]
|-
| BB(14)
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| Erdős Ternary Conjecture Variant
| Halts iff the following conjecture (a variant of #406) is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1."
| BB(4,4)
| BBi(15)
| {{TM|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
| Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| Sloane Ternary Conjecture
| Halts if and only if the following [http://neilsloane.com/doc/persistence.html conjecture by N. J. A. Sloane] is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n."
| BB(6,4)
| BBi(21)
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
| Not independently verified.
|-
| rowspan="2"|Weak Collatz
| rowspan="2"|Halts iff the weak Collatz conjecture is false.
|BB(124) <br> BB(43,4)
|BBi(121)
|style="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)
|style="white-space: nowrap; text-align: right;"|Jul 2021
|[[User:Cosmo|Tristan Stérin]]
|The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). Not independently verified, and probably easy to further optimise.
|-
|BB(14) <br> BB(10,3)
|BBi(26)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz (includes Lean formalization)
{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|[https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| Odd multiply perfect
| Halts iff an odd [[wikipedia:Multiply perfect number|multiply perfect]] greater than 1 exists.
|BB(23,3)
|BBi(63)
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
| Odd perfect
| Halts iff an odd [[wikipedia:Perfect number|perfect]] exists.
|BB(37)
|BBi(71)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Not independently verified.
|-
|Sierpiński
|Halts iff 78,557 is the smallest [[wikipedia:Sierpiński number|Sierpiński number]].
|BB(60)
|BBi(104)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Not independently verified.
|-
|Riesel
|Halts iff 509,203 is the smallest [[wikipedia:Riesel number|Riesel number]].
|BB(68)
|BBi(112)
|style="word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|Not independently verified.
|-
| rowspan="2"|Fermat's Last Theorem
| rowspan="2"|Does not halt
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). The machine was a cryptid until Wiles resolved Fermat's Last Theorem in 1994; it is now known that it cannot halt. 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.
|-
|BB(90)
|BBi(160)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|Not independently verified.
|-
| Bigfoot - compiled
| Halts iff [[Bigfoot]] halts.
| [[BB(7)]]
| BBi(13)
|style="word-break:break-word"| {{TM|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| Hydra - compiled
| Halts iff [[Hydra]] halts.
|BB(9)
|BBi(15)
|style="word-break:break-word"|{{TM|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}
[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|-
| Antihydra-like
|
|[[BB(7)]]
|BBi(13)
|style="word-break:break-word"| {{TM|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|2n -> 3n, 2n+1 -> 3n+5, start at 2, halts only when the number of odd numbers is more than twice the number of even numbers.
|}

== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|https://github.com/calr3/tinycryptids/tree/main/twinprimes
{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T14:39:01Z

Frogmatic: /* Larger Cryptids */ Give exact month of the original Goldbach machine (according to the github date)

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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 !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF consistency]]
|BB(432)
|
|style="word-break:break-word"|[https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e Wade's machine]
|style="white-space: nowrap; text-align: right;"|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.
|-
|ZFC consistency
|BB(215)
|
|style="word-break:break-word"|https://github.com/elliotglazer/k215-pa
|style="white-space: nowrap; text-align: right;"|Oct 2026
|elliotglazer
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent, machine-checked in Peano-Arithmetic. Not independently verified.
|-
|Peano Arithmetic consistency
|BB(372)
|BBi(742)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent. [https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | Riemann Hypothesis
|BB(744)
|
|style="word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(120)
|BBi(238)
|style="word-break:break-word"|https://arxiv.org/abs/2609.30306
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(95)
|BBi(189)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false. Not independently verified.
|-
| rowspan="2"|Goldbach
|BB(25)
|BBi(49)
|style="word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
{{WrapTM|Machine code|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|undecided}}
|style="white-space: nowrap; text-align: right;"|May 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>
|-
|BB(21)
|BBi(41)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Not independently verified.
|-
| rowspan="2"| Erdős Ternary Conjecture
| BB(15) <br> BB(5,4)
| BBi(19)
|style="word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
{{WrapTM|BB(5,4) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|style="white-space: nowrap; text-align: right;"|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." [https://arxiv.org/abs/2107.12475 arxiv preprint] [https://www.erdosproblems.com/406 Erdős Problems]
|-
| BB(14)
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|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." Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| Erdős Ternary Conjecture Variant
| BB(4,4)
| BBi(15)
| {{TM|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
|The machine halts if and only if the following conjecture is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1." Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| Sloane Ternary Conjecture
| BB(6,4)
| BBi(21)
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if the following conjecture by N. J. A. Sloane is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n." Not independently verified. [http://neilsloane.com/doc/persistence.html Sloane's Page]
|-
| rowspan="2"|Weak Collatz
|BB(124) <br> BB(43,4)
|BBi(121)
|style="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)
|style="white-space: nowrap; text-align: right;"|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.
|-
|BB(14) <br> BB(10,3)
|BBi(26)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz (includes Lean formalization)
{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if the "weak Collatz conjecture" is false, as above. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| rowspan="2"|Odd Perfect
|BB(23,3)
|BBi(63)
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|The machine halts if and only if there exists an odd multiply perfect number greater than 1.
Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
|BB(37)
|BBi(71)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if there exists an odd perfect number. Not independently verified.
|-
|Sierpiński
|BB(60)
|BBi(104)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if 78557 is the smallest Sierpiński number. Not independently verified.
|-
|Riesel
|BB(68)
|BBi(112)
|style="word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if 509203 is the smallest Riesel number. Not independently verified.
|-
| rowspan="2"|Fermat's Last Theorem
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). 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.
|-
|BB(90)
|BBi(160)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if Fermat's Last Theorem is false, so it never halts. Not independently verified.
|-
| Bigfoot - compiled
| [[BB(7)]]
| BBi(13)
|style="word-break:break-word"| {{TM|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| Hydra - compiled
|BB(9)
|BBi(15)
|style="word-break:break-word"|{{TM|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}

[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|-
| Antihydra-like
|[[BB(7)]]
|BBi(13)
|style="word-break:break-word"| {{TM|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|2n -> 3n, 2n+1 -> 3n+5, start at 2, halts only when the number of odd numbers is more than twice the number of even numbers.
|}

== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|https://github.com/calr3/tinycryptids/tree/main/twinprimes
{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T14:35:46Z

Frogmatic: /* Larger Cryptids */ Fix outdated Metamath Github link

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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 !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF consistency]]
|BB(432)
|
|style="word-break:break-word"|[https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e Wade's machine]
|style="white-space: nowrap; text-align: right;"|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.
|-
|ZFC consistency
|BB(215)
|
|style="word-break:break-word"|https://github.com/elliotglazer/k215-pa
|style="white-space: nowrap; text-align: right;"|Oct 2026
|elliotglazer
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent, machine-checked in Peano-Arithmetic. Not independently verified.
|-
|Peano Arithmetic consistency
|BB(372)
|BBi(742)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent. [https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | Riemann Hypothesis
|BB(744)
|
|style="word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(120)
|BBi(238)
|style="word-break:break-word"|https://arxiv.org/abs/2609.30306
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(95)
|BBi(189)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false. Not independently verified.
|-
| rowspan="2"|Goldbach
|BB(25)
|BBi(49)
|style="word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
{{WrapTM|Machine code|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|undecided}}
|style="white-space: nowrap; text-align: right;"|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>
|-
|BB(21)
|BBi(41)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Not independently verified.
|-
| rowspan="2"| Erdős Ternary Conjecture
| BB(15) <br> BB(5,4)
| BBi(19)
|style="word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
{{WrapTM|BB(5,4) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|style="white-space: nowrap; text-align: right;"|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." [https://arxiv.org/abs/2107.12475 arxiv preprint] [https://www.erdosproblems.com/406 Erdős Problems]
|-
| BB(14)
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|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." Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| Erdős Ternary Conjecture Variant
| BB(4,4)
| BBi(15)
| {{TM|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
|The machine halts if and only if the following conjecture is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1." Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| Sloane Ternary Conjecture
| BB(6,4)
| BBi(21)
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if the following conjecture by N. J. A. Sloane is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n." Not independently verified. [http://neilsloane.com/doc/persistence.html Sloane's Page]
|-
| rowspan="2"|Weak Collatz
|BB(124) <br> BB(43,4)
|BBi(121)
|style="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)
|style="white-space: nowrap; text-align: right;"|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.
|-
|BB(14) <br> BB(10,3)
|BBi(26)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz (includes Lean formalization)
{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if the "weak Collatz conjecture" is false, as above. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| rowspan="2"|Odd Perfect
|BB(23,3)
|BBi(63)
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|The machine halts if and only if there exists an odd multiply perfect number greater than 1.
Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
|BB(37)
|BBi(71)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if there exists an odd perfect number. Not independently verified.
|-
|Sierpiński
|BB(60)
|BBi(104)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if 78557 is the smallest Sierpiński number. Not independently verified.
|-
|Riesel
|BB(68)
|BBi(112)
|style="word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if 509203 is the smallest Riesel number. Not independently verified.
|-
| rowspan="2"|Fermat's Last Theorem
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). 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.
|-
|BB(90)
|BBi(160)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if Fermat's Last Theorem is false, so it never halts. Not independently verified.
|-
| Bigfoot - compiled
| [[BB(7)]]
| BBi(13)
|style="word-break:break-word"| {{TM|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| Hydra - compiled
|BB(9)
|BBi(15)
|style="word-break:break-word"|{{TM|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}

[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|-
| Antihydra-like
|[[BB(7)]]
|BBi(13)
|style="word-break:break-word"| {{TM|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|2n -> 3n, 2n+1 -> 3n+5, start at 2, halts only when the number of odd numbers is more than twice the number of even numbers.
|}

== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|https://github.com/calr3/tinycryptids/tree/main/twinprimes
{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T14:34:07Z

Frogmatic: Use WrapTM more consistently

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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 !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF consistency]]
|BB(432)
|
|style="word-break:break-word"|[https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e Wade's machine]
|style="white-space: nowrap; text-align: right;"|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.
|-
|ZFC consistency
|BB(215)
|
|style="word-break:break-word"|https://github.com/elliotglazer/k215-pa
|style="white-space: nowrap; text-align: right;"|Oct 2026
|elliotglazer
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent, machine-checked in Peano-Arithmetic. Not independently verified.
|-
|Peano Arithmetic consistency
|BB(372)
|BBi(742)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent. [https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | Riemann Hypothesis
|BB(744)
|
|style="word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/riemann-matiyasevich-aaronson.nql
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(120)
|BBi(238)
|style="word-break:break-word"|https://arxiv.org/abs/2609.30306
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(95)
|BBi(189)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false. Not independently verified.
|-
| rowspan="2"|Goldbach
|BB(25)
|BBi(49)
|style="word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
{{WrapTM|Machine code|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|undecided}}
|style="white-space: nowrap; text-align: right;"|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>
|-
|BB(21)
|BBi(41)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Not independently verified.
|-
| rowspan="2"| Erdős Ternary Conjecture
| BB(15) <br> BB(5,4)
| BBi(19)
|style="word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
{{WrapTM|BB(5,4) machine|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|style="white-space: nowrap; text-align: right;"|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." [https://arxiv.org/abs/2107.12475 arxiv preprint] [https://www.erdosproblems.com/406 Erdős Problems]
|-
| BB(14)
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|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." Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| Erdős Ternary Conjecture Variant
| BB(4,4)
| BBi(15)
| {{TM|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
|The machine halts if and only if the following conjecture is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1." Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| Sloane Ternary Conjecture
| BB(6,4)
| BBi(21)
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if the following conjecture by N. J. A. Sloane is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n." Not independently verified. [http://neilsloane.com/doc/persistence.html Sloane's Page]
|-
| rowspan="2"|Weak Collatz
|BB(124) <br> BB(43,4)
|BBi(121)
|style="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)
|style="white-space: nowrap; text-align: right;"|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.
|-
|BB(14) <br> BB(10,3)
|BBi(26)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz (includes Lean formalization)
{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if the "weak Collatz conjecture" is false, as above. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| rowspan="2"|Odd Perfect
|BB(23,3)
|BBi(63)
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|The machine halts if and only if there exists an odd multiply perfect number greater than 1.
Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
|BB(37)
|BBi(71)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if there exists an odd perfect number. Not independently verified.
|-
|Sierpiński
|BB(60)
|BBi(104)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if 78557 is the smallest Sierpiński number. Not independently verified.
|-
|Riesel
|BB(68)
|BBi(112)
|style="word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if 509203 is the smallest Riesel number. Not independently verified.
|-
| rowspan="2"|Fermat's Last Theorem
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). 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.
|-
|BB(90)
|BBi(160)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if Fermat's Last Theorem is false, so it never halts. Not independently verified.
|-
| Bigfoot - compiled
| [[BB(7)]]
| BBi(13)
|style="word-break:break-word"| {{TM|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| Hydra - compiled
|BB(9)
|BBi(15)
|style="word-break:break-word"|{{TM|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}

[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|-
| Antihydra-like
|[[BB(7)]]
|BBi(13)
|style="word-break:break-word"| {{TM|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Jacobzheng
|2n -> 3n, 2n+1 -> 3n+5, start at 2, halts only when the number of odd numbers is more than twice the number of even numbers.
|}

== 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|https://github.com/calr3/tinycryptids/tree/main/twinprimes
{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T14:20:05Z

Frogmatic: Improve formatting of Date column

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Mar 2025
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|<code>mxdys</code>
|Variant of Hydra and Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Sep 2024
|Daniel Yuan
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Jan 2025
|<code>mxdys</code>
|Probviously non-halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}
|style="white-space: nowrap; text-align: right;"|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|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|style="white-space: nowrap; text-align: right;"|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|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}}
|style="white-space: nowrap; text-align: right;"|Apr 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}}
|style="white-space: nowrap; text-align: right;"|Feb 2025
|Racheline
|Probviously 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 !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF consistency]]
|BB(432)
|
|style="word-break:break-word"|[https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e Wade's machine]
|style="white-space: nowrap; text-align: right;"|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.
|-
|ZFC consistency
|BB(215)
|
|style="word-break:break-word"|https://github.com/elliotglazer/k215-pa
|style="white-space: nowrap; text-align: right;"|Oct 2026
|elliotglazer
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent, machine-checked in Peano-Arithmetic. Not independently verified.
|-
|Peano Arithmetic consistency
|BB(372)
|BBi(742)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|style="white-space: nowrap; text-align: right;"|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent. [https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | Riemann Hypothesis
|BB(744)
|
|style="word-break:break-word"|https://github.com/sorear/metamath-turing-machines/blob/master/riemann-matiyasevich-aaronson.nql
|style="white-space: nowrap; text-align: right;"|2016
|Matiyasevich and O’Rear
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(120)
|BBi(238)
|style="word-break:break-word"|https://arxiv.org/abs/2609.30306
|style="white-space: nowrap; text-align: right;"|Sep 2026
|Joseph M. Shunia
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(95)
|BBi(189)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false. Not independently verified.
|-
| rowspan="2"|Goldbach
|BB(25)
|BBi(49)
|style="word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
{{WrapTM|Machine code|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|undecided}}
|style="white-space: nowrap; text-align: right;"|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>
|-
|BB(21)
|BBi(41)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|Google DeepMind
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Not independently verified.
|-
| rowspan="2"| Erdős Ternary Conjecture
| BB(15) <br> BB(5,4)
| BBi(19)
|style="word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
{{TM|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
|style="white-space: nowrap; text-align: right;"|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." [https://arxiv.org/abs/2107.12475 arxiv preprint] [https://www.erdosproblems.com/406 Erdős Problems]
|-
| BB(14)
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|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." Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| Erdős Ternary Conjecture Variant
| BB(4,4)
| BBi(15)
| {{TM|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|roromachine14
|The machine halts if and only if the following conjecture is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1." Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| Sloane Ternary Conjecture
| BB(6,4)
| BBi(21)
|style="word-break:break-all"| {{TM|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if the following conjecture by N. J. A. Sloane is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n." Not independently verified. [http://neilsloane.com/doc/persistence.html Sloane's Page]
|-
| rowspan="2"|Weak Collatz
|BB(124) <br> BB(43,4)
|BBi(121)
|style="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)
|style="white-space: nowrap; text-align: right;"|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.
|-
|BB(14) <br> BB(10,3)
|BBi(26)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz (includes Lean formalization)
{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if the "weak Collatz conjecture" is false, as above. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| rowspan="2"|Odd Perfect
|BB(23,3)
|BBi(63)
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|kg583
|The machine halts if and only if there exists an odd multiply perfect number greater than 1.
Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
|BB(37)
|BBi(71)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if there exists an odd perfect number. Not independently verified.
|-
|Sierpiński
|BB(60)
|BBi(104)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if 78557 is the smallest Sierpiński number. Not independently verified.
|-
|Riesel
|BB(68)
|BBi(112)
|style="word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if 509203 is the smallest Riesel number. Not independently verified.
|-
| rowspan="2"|Fermat's Last Theorem
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|style="white-space: nowrap; text-align: right;"|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). 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.
|-
|BB(90)
|BBi(160)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat (includes Lean formalization)
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if Fermat's Last Theorem is false, so it never halts. Not independently verified.
|-
| Bigfoot - compiled
| [[BB(7)]]
| BBi(13)
|style="word-break:break-word"| {{TM|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| Hydra - compiled
|BB(9)
|BBi(15)
|style="word-break:break-word"|{{TM|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}

[[File:Hydra_9_states.txt]]
|style="white-space: nowrap; text-align: right;"|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|style="white-space: nowrap; text-align: right;"|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|https://github.com/calr3/tinycryptids/tree/main/twinprimes
{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Fenrir|Fenrir]]
|[[Fractran|BBf(22)]]
|<code>[1/15, 27/77, 49/3, 10/49, 33/2]</code> and 2 others
|style="white-space: nowrap; text-align: right;"|Mar 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|style="white-space: nowrap; text-align: right;"|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|style="white-space: nowrap; text-align: right;"|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|style="white-space: nowrap; text-align: right;"|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|style="white-space: nowrap; text-align: right;"|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|style="white-space: nowrap; text-align: right;"|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|style="white-space: nowrap; text-align: right;"|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|style="white-space: nowrap; text-align: right;"|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|style="white-space: nowrap; text-align: right;"|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|style="white-space: nowrap; text-align: right;"|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|style="white-space: nowrap; text-align: right;"|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|style="white-space: nowrap; text-align: right;"|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T14:15:04Z

Frogmatic: Use better Wikipedia links (now I know how to do that!)

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|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|undecided}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|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}}
|Apr 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.
|}
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 !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF consistency]]
|BB(432)
|
|style="word-break:break-word"|[https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e Wade's machine]
|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.
|-
|ZFC consistency
|BB(215)
|
|style="word-break:break-word"|https://github.com/elliotglazer/k215-pa
|Oct 2026
|elliotglazer
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent, machine-checked in Peano-Arithmetic. Not independently verified.
|-
|Peano Arithmetic consistency
|BB(372)
|BBi(742)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent. [https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | Riemann Hypothesis
|BB(744)
|
|style="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 the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(120)
|BBi(238)
|style="word-break:break-word"|https://arxiv.org/abs/2609.30306
|Sep 2026
|Joseph M. Shunia
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(95)
|BBi(189)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|Oct 2026
|Google DeepMind
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false. Not independently verified.
|-
| rowspan="2"|Goldbach
|BB(25)
|BBi(49)
|style="word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
{{WrapTM|Machine code|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|undecided}}
|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>
|-
|BB(21)
|BBi(41)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|Oct 2026
|Google DeepMind
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Not independently verified.
|-
| rowspan="2"| Erdős Ternary Conjecture
| BB(15) <br> BB(5,4)
| BBi(19)
|style="word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
{{WrapTM|BB(15) machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA|undecided}}
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
{{TM|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE|undecided}}
| 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." [https://arxiv.org/abs/2107.12475 arxiv preprint] [https://www.erdosproblems.com/406 Erdős Problems]
|-
| BB(14)
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|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." Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| Erdős Ternary Conjecture Variant
| BB(4,4)
| BBi(15)
| {{TM|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|Oct 2026
|roromachine14
|The machine halts if and only if the following conjecture is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1." Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| Sloane Ternary Conjecture
| BB(6,4)
| BBi(21)
|style="word-break:break-all"| {{TM|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if the following conjecture by N. J. A. Sloane is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n." Not independently verified. [http://neilsloane.com/doc/persistence.html Sloane's Page]
|-
| rowspan="2"|Weak Collatz
|BB(124) <br> BB(43,4)
|BBi(121)
|style="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.
|-
|BB(14) <br> BB(10,3)
|BBi(26)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz (includes Lean formalization)
{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(10,3) machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if the "weak Collatz conjecture" is false, as above. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| rowspan="2"|Odd Perfect
|BB(23,3)
|BBi(63)
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|Jul 2024
|kg583
|The machine halts if and only if there exists an odd multiply perfect number greater than 1.
Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
|BB(37)
|BBi(71)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if there exists an odd perfect number. Not independently verified.
|-
|Sierpiński
|BB(60)
|BBi(104)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if 78557 is the smallest Sierpiński number. Not independently verified.
|-
|Riesel
|BB(68)
|BBi(112)
|style="word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel (includes Lean formalization)
|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if 509203 is the smallest Riesel number. Not independently verified.
|-
| rowspan="2"|Fermat's Last Theorem
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). 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.
|-
|BB(90)
|BBi(160)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if Fermat's Last Theorem is false, so it never halts. Not independently verified.
|-
| Bigfoot - compiled
| [[BB(7)]]
| BBi(13)
|style="word-break:break-word"| {{TM|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| Hydra - compiled
|BB(9)
|BBi(15)
|style="word-break:break-word"|{{TM|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}

[[File:Hydra_9_states.txt]]
|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[[wikipedia:Twin prime#Twin_prime_conjecture|Twin prime conjecture]]
|[[Beeping Busy Beaver|BBB(18)]]
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/twinprimes
{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|style="width:30%;word-break:break-word"|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [[wikipedia:Collatz conjecture|Collatz conjecture]] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[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
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T14:11:51Z

Frogmatic: Further tidy-up of machine code formatting to improve table readability

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|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|undecided}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|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}}
|Apr 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.
|}
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 !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF consistency]]
|BB(432)
|
|style="word-break:break-word"|[https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e Wade's machine]
|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.
|-
|ZFC consistency
|BB(215)
|
|style="word-break:break-word"|https://github.com/elliotglazer/k215-pa
|Oct 2026
|elliotglazer
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent, machine-checked in Peano-Arithmetic. Not independently verified.
|-
|Peano Arithmetic consistency
|BB(372)
|BBi(742)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent. [https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | Riemann Hypothesis
|BB(744)
|
|style="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 the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(120)
|BBi(238)
|style="word-break:break-word"|https://arxiv.org/abs/2609.30306
|Sep 2026
|Joseph M. Shunia
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(95)
|BBi(189)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|Oct 2026
|Google DeepMind
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false. Not independently verified.
|-
| rowspan="2"|Goldbach
|BB(25)
|BBi(49)
|style="word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
{{WrapTM|Machine code|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|undecided}}
|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>
|-
|BB(21)
|BBi(41)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|Oct 2026
|Google DeepMind
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Not independently verified.
|-
| rowspan="2"| Erdős Ternary Conjecture
| BB(15) <br> BB(5,4)
| BBi(19)
|style="word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_5_4.txt
{{WrapTM|4-state machine code|1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE}}
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
{{WrapTM|2-state machine|1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA}}
| 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." [https://arxiv.org/abs/2107.12475 arxiv preprint] [https://www.erdosproblems.com/406 Erdős Problems]
|-
| BB(14)
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|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." Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| Erdős Ternary Conjecture Variant
| BB(4,4)
| BBi(15)
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|Oct 2026
|roromachine14
|The machine halts if and only if the following conjecture is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1." Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| Sloane Ternary Conjecture
| BB(6,4)
| BBi(21)
|style="word-break:break-all"| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if the following conjecture by N. J. A. Sloane is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n." Not independently verified. [http://neilsloane.com/doc/persistence.html Sloane's Page]
|-
| rowspan="2"|Weak Collatz
|BB(124) <br> BB(43,4)
|BBi(121)
|style="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.
|-
|BB(14) <br> BB(10,3)
|BBi(26)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz (includes Lean formalization)
{{WrapTM|2-state machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|3-state machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if the "weak Collatz conjecture" is false, as above. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| rowspan="2"|Odd Perfect
|BB(23,3)
|BBi(63)
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|Jul 2024
|kg583
|The machine halts if and only if there exists an odd multiply perfect number greater than 1.
Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
|BB(37)
|BBi(71)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if there exists an odd perfect number. Not independently verified.
|-
|Sierpiński
|BB(60)
|BBi(104)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if 78557 is the smallest Sierpiński number. Not independently verified.
|-
|Riesel
|BB(68)
|BBi(112)
|style="word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel (includes Lean formalization)
|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if 509203 is the smallest Riesel number. Not independently verified.
|-
| rowspan="2"|Fermat's Last Theorem
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). 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.
|-
|BB(90)
|BBi(160)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if Fermat's Last Theorem is false, so it never halts. Not independently verified.
|-
| Bigfoot - compiled
| [[BB(7)]]
| BBi(13)
|style="word-break:break-word"| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| Hydra - compiled
|BB(9)
|BBi(15)
|style="word-break:break-word"|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}

[[File:Hydra_9_states.txt]]
|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[https://en.wikipedia.org/wiki/Twin_prime#Twin_prime_conjecture Twin Prime]
|[[Beeping Busy Beaver|BBB(18)]]
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/twinprimes
{{WrapTM|Machine code (beep state: N)|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|style="width:30%;word-break:break-word"|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
{{WrapTM|Machine code (beep state: O)|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}
|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [https://en.wikipedia.org/wiki/Collatz_conjecture Collatz conjecture] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[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
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T14:04:04Z

Frogmatic: /* Larger Cryptids */ Use WrapTM template so the styling of machine formatting stays consistent

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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|undecided}}
|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|undecided}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB|undecided}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA|undecided}}
|Jul 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE|undecided}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA|undecided}}
|Nov 2024
|Racheline
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE|undecided}}
|Jul 2025
|<code>mxdys</code>
|Probviously halting.
|-
|
|[[BB(6)]]
|{{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE|undecided}}
|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}}
|Apr 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.
|}
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 !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF consistency]]
|BB(432)
|
|style="word-break:break-word"|[https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e Wade's machine]
|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.
|-
|ZFC consistency
|BB(215)
|
|style="word-break:break-word"|https://github.com/elliotglazer/k215-pa
|Oct 2026
|elliotglazer
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent, machine-checked in Peano-Arithmetic. Not independently verified.
|-
|Peano Arithmetic consistency
|BB(372)
|BBi(742)
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent. [https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | Riemann Hypothesis
|BB(744)
|
|style="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 the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(120)
|BBi(238)
|style="word-break:break-word"|https://arxiv.org/abs/2609.30306
|Sep 2026
|Joseph M. Shunia
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(95)
|BBi(189)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|Oct 2026
|Google DeepMind
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false. Not independently verified.
|-
| rowspan="2"|Goldbach
|BB(25)
|BBi(49)
|style="word-break:break-word"|https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
{{WrapTM|Machine code|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|undecided}}
|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>
|-
|BB(21)
|BBi(41)
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
|Oct 2026
|Google DeepMind
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Not independently verified.
|-
| rowspan="2"| Erdős Ternary Conjecture
| BB(15) <br> BB(5,4)
| BBi(19)
|style="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>
| 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." [https://arxiv.org/abs/2107.12475 arxiv preprint] [https://www.erdosproblems.com/406 Erdős Problems]
|-
| BB(14)
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|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." Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| Erdős Ternary Conjecture Variant
| BB(4,4)
| BBi(15)
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|Oct 2026
|roromachine14
|The machine halts if and only if the following conjecture is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1." Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| Sloane Ternary Conjecture
| BB(6,4)
| BBi(21)
|style="word-break:break-all"| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if the following conjecture by N. J. A. Sloane is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n." Not independently verified. [http://neilsloane.com/doc/persistence.html Sloane's Page]
|-
| rowspan="2"|Weak Collatz
|BB(124) <br> BB(43,4)
|BBi(121)
|style="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.
|-
|BB(14) <br> BB(10,3)
|BBi(26)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz (includes Lean formalization)
{{WrapTM|2-state machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|3-state machine code|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if the "weak Collatz conjecture" is false, as above. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| rowspan="2"|Odd Perfect
|BB(23,3)
|BBi(63)
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|Jul 2024
|kg583
|The machine halts if and only if there exists an odd multiply perfect number greater than 1.
Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
|BB(37)
|BBi(71)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if there exists an odd perfect number. Not independently verified.
|-
|Sierpiński
|BB(60)
|BBi(104)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if 78557 is the smallest Sierpiński number. Not independently verified.
|-
|Riesel
|BB(68)
|BBi(112)
|style="word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel (includes Lean formalization)
|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if 509203 is the smallest Riesel number. Not independently verified.
|-
| rowspan="2"|Fermat's Last Theorem
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). 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.
|-
|BB(90)
|BBi(160)
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if Fermat's Last Theorem is false, so it never halts. Not independently verified.
|-
| Bigfoot - compiled
| [[BB(7)]]
| BBi(13)
|style="word-break:break-word"| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| Hydra - compiled
|BB(9)
|BBi(15)
|style="word-break:break-word"|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}

[[File:Hydra_9_states.txt]]
|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[https://en.wikipedia.org/wiki/Twin_prime#Twin_prime_conjecture Twin Prime]
|[[Beeping Busy Beaver|BBB(18)]]
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/twinprimes
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}</div><br>Beeping State: N</div></div>
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|style="width:30%;word-break:break-word"|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}</div><br>Beeping State: O</div></div>
|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [https://en.wikipedia.org/wiki/Collatz_conjecture Collatz conjecture] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[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
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts if and only if 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|Oct 2026
|sheep
|This machine halts if and only if 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|These Wang tiles tile the whole integer plane if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is consistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Template:WrapTM

2026-10-25T13:57:03Z

Frogmatic: Add full wrapper

<div class="toccolours mw-collapsible mw-collapsed">{{{1}}}<div class="mw-collapsible-content" style="word-break:break-all">{{TM|{{{2}}}|{{{3}}}}}</div></div><noinclude>
A collapsible wrapper around a fully-specified Turing Machine, for convenient use where space is limited.
</noinclude>

Template:WrapTM

2026-10-25T13:54:19Z

Frogmatic: Create a wrapping template for embedding the {{TM}} template in a collapse way.

<div class="mw-collapsible-content" style="word-break:break-all">{{TM|{{{1}}}|{{{2}}}}}</div><noinclude>
A collapsible wrapper around a fully-specified Turing Machine, for convenient use where space is limited.
</noinclude>

Cryptids

2026-10-25T13:19:03Z

Frogmatic: /* Larger Cryptids */ More descriptive names for initialisms

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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}}
|Apr 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.
|}
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 !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
|-
|[[Logical independence|ZF consistency]]
|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.
|-
|ZFC consistency
|BB(215)
|
|style="width:30%;word-break:break-word"|https://github.com/elliotglazer/k215-pa
|Oct 2026
|elliotglazer
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is inconsistent, machine-checked in Peano-Arithmetic. Not independently verified.
|-
|Peano Arithmetic consistency
|BB(372)
|
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent. [https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | Riemann Hypothesis
|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 the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(120)
|BBi(238)
|style="width:30%;word-break:break-word"|https://arxiv.org/abs/2609.30306
|Sep 2026
|Joseph M. Shunia
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(95)
|BBi(189)
|style="width:30%;word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|Oct 2026
|Google DeepMind
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false. Not independently verified.
|-
| rowspan="2"|Goldbach
|BB(25)
|BBi(49)
|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"><div style="word-break:break-all">{{TM|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|undecided}}</div></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>
|-
|BB(21)
|BBi(41)
|style="width:30%;word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}</div></div></div>
|Oct 2026
|Google DeepMind
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Not independently verified.
|-
| rowspan="2"| Erdős Ternary Conjecture
| BB(15) <br> BB(5,4)
| BBi(19)
|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>
| 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." [https://arxiv.org/abs/2107.12475 arxiv preprint] [https://www.erdosproblems.com/406 Erdős Problems]
|-
| BB(14)
| BBi(27)
| <div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}</div></div></div>
|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|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." Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| Erdős Ternary Conjecture Variant
| BB(4,4)
| BBi(15)
| {{TM|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|Oct 2026
|roromachine14
|The machine halts if and only if the following conjecture is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1." Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| Sloane Ternary Conjecture
| BB(6,4)
| BBi(21)
|style="width:30%;word-break:break-all"| {{TM|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if the following conjecture by N. J. A. Sloane is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n." Not independently verified. [http://neilsloane.com/doc/persistence.html Sloane's Page]
|-
| rowspan="2"|Weak Collatz
|BB(124) <br> BB(43,4)
|BBi(121)
|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.
|-
|BB(14) <br> BB(10,3)
|BBi(26)
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz (includes Lean formalization)
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}<br>{{TM|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}</div></div></div>
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if the "weak Collatz conjecture" is false, as above. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| rowspan="2"|Odd Perfect
|BB(23,3)
|BBi(63)
|<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}</div></div></div>
|Jul 2024
|kg583
|The machine halts if and only if there exists an odd multiply perfect number greater than 1.
Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
|BB(37)
|BBi(71)
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if there exists an odd perfect number. Not independently verified.
|-
|Sierpiński
|BB(60)
|BBi(104)
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if 78557 is the smallest Sierpiński number. Not independently verified.
|-
|Riesel
|BB(68)
|BBi(112)
|style="width:30%;word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel (includes Lean formalization)
|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if 509203 is the smallest Riesel number. Not independently verified.
|-
| rowspan="2"|Fermat's Last Theorem
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). 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.
|-
|BB(90)
|BBi(160)
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if Fermat's Last Theorem is false, so it never halts. Not independently verified.
|-
| Bigfoot - compiled
| [[BB(7)]]
| BBi(13)
|style="width:30%;word-break:break-word"| {{TM|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| Hydra - compiled
|BB(9)
|BBi(15)
|style="width:30%;word-break:break-word"|{{TM|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}

[[File:Hydra_9_states.txt]]
|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[https://en.wikipedia.org/wiki/Twin_prime#Twin_prime_conjecture Twin Prime]
|[[Beeping Busy Beaver|BBB(18)]]
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/twinprimes
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}</div><br>Beeping State: N</div></div>
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|style="width:30%;word-break:break-word"|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}</div><br>Beeping State: O</div></div>
|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [https://en.wikipedia.org/wiki/Collatz_conjecture Collatz conjecture] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[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
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T13:17:42Z

Frogmatic: /* Beeping Busy Beaver */ Context link for Twin prime conjecture

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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}}
|Apr 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.
|}
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 !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! 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.
|-
|ZFC
|BB(215)
|
|style="width:30%;word-break:break-word"|https://github.com/elliotglazer/k215-pa
|Oct 2026
|elliotglazer
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is inconsistent, machine-checked in Peano-Arithmetic. Not independently verified.
|-
|PA
|BB(372)
|
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent. [https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | 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 the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(120)
|BBi(238)
|style="width:30%;word-break:break-word"|https://arxiv.org/abs/2609.30306
|Sep 2026
|Joseph M. Shunia
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(95)
|BBi(189)
|style="width:30%;word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|Oct 2026
|Google DeepMind
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false. Not independently verified.
|-
| rowspan="2"|Goldbach
|BB(25)
|BBi(49)
|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"><div style="word-break:break-all">{{TM|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|undecided}}</div></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>
|-
|BB(21)
|BBi(41)
|style="width:30%;word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}</div></div></div>
|Oct 2026
|Google DeepMind
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Not independently verified.
|-
| rowspan="2"| Erdős Ternary Conjecture
| BB(15) <br> BB(5,4)
| BBi(19)
|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>
| 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." [https://arxiv.org/abs/2107.12475 arxiv preprint] [https://www.erdosproblems.com/406 Erdős Problems]
|-
| BB(14)
| BBi(27)
| <div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}</div></div></div>
|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|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." Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| Erdős Ternary Conjecture Variant
| BB(4,4)
| BBi(15)
| {{TM|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|Oct 2026
|roromachine14
|The machine halts if and only if the following conjecture is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1." Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| Sloane Ternary Conjecture
| BB(6,4)
| BBi(21)
|style="width:30%;word-break:break-all"| {{TM|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if the following conjecture by N. J. A. Sloane is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n." Not independently verified. [http://neilsloane.com/doc/persistence.html Sloane's Page]
|-
| rowspan="2"|Weak Collatz
|BB(124) <br> BB(43,4)
|BBi(121)
|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.
|-
|BB(14) <br> BB(10,3)
|BBi(26)
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz (includes Lean formalization)
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}<br>{{TM|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}</div></div></div>
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if the "weak Collatz conjecture" is false, as above. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| rowspan="2"|Odd Perfect
|BB(23,3)
|BBi(63)
|<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}</div></div></div>
|Jul 2024
|kg583
|The machine halts if and only if there exists an odd multiply perfect number greater than 1.
Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
|BB(37)
|BBi(71)
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if there exists an odd perfect number. Not independently verified.
|-
|Sierpiński
|BB(60)
|BBi(104)
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if 78557 is the smallest Sierpiński number. Not independently verified.
|-
|Riesel
|BB(68)
|BBi(112)
|style="width:30%;word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel (includes Lean formalization)
|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if 509203 is the smallest Riesel number. Not independently verified.
|-
| rowspan="2"|Fermat's Last Theorem
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). 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.
|-
|BB(90)
|BBi(160)
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if Fermat's Last Theorem is false, so it never halts. Not independently verified.
|-
| Bigfoot - compiled
| [[BB(7)]]
| BBi(13)
|style="width:30%;word-break:break-word"| {{TM|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| Hydra - compiled
|BB(9)
|BBi(15)
|style="width:30%;word-break:break-word"|{{TM|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}

[[File:Hydra_9_states.txt]]
|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|[https://en.wikipedia.org/wiki/Twin_prime#Twin_prime_conjecture Twin Prime]
|[[Beeping Busy Beaver|BBB(18)]]
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/twinprimes
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}</div><br>Beeping State: N</div></div>
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|style="width:30%;word-break:break-word"|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}</div><br>Beeping State: O</div></div>
|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [https://en.wikipedia.org/wiki/Collatz_conjecture Collatz conjecture] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[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
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T13:15:14Z

Frogmatic: Make all rows consistently one cell per line

[[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/BBi domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Bigfoot]]
|[[BB(3,3)]] [[Instruction-Limited Busy Beaver|BBi(8)]]
|{{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}
|Nov 2023
|[[User:Sligocki|Shawn Ligocki]]
|Probviously non-halting.
|-
|[[Hydra]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Bonus cryptid]]
|[[BB(2,5)]] [[Instruction-Limited Busy Beaver|BBi(9)]]
|{{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}
|May 2024
|Daniel Yuan
|Probviously non-halting.
|-
|[[Antihydra]]
|[[BB(6)]]
|{{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA|undecided}}
|Jun 2024
|<code>mxdys</code>, shown to be a Cryptid by Racheline.
|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}}
|Apr 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.
|}
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 !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! 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.
|-
|ZFC
|BB(215)
|
|style="width:30%;word-break:break-word"|https://github.com/elliotglazer/k215-pa
|Oct 2026
|elliotglazer
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] with the Axiom of Choice is inconsistent, machine-checked in Peano-Arithmetic. Not independently verified.
|-
|PA
|BB(372)
|
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|2026
|LegionMammal
|The machines halts if and only if Peano-Arithmetic is inconsistent. [https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
| rowspan="3" | 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 the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(120)
|BBi(238)
|style="width:30%;word-break:break-word"|https://arxiv.org/abs/2609.30306
|Sep 2026
|Joseph M. Shunia
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|BB(95)
|BBi(189)
|style="width:30%;word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|Oct 2026
|Google DeepMind
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false. Not independently verified.
|-
| rowspan="2"|Goldbach
|BB(25)
|BBi(49)
|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"><div style="word-break:break-all">{{TM|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|undecided}}</div></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>
|-
|BB(21)
|BBi(41)
|style="width:30%;word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}</div></div></div>
|Oct 2026
|Google DeepMind
|The machine halts if and only if [[wikipedia:Goldbach's_conjecture|Goldbach's conjecture]] is false. Not independently verified.
|-
| rowspan="2"| Erdős Ternary Conjecture
| BB(15) <br> BB(5,4)
| BBi(19)
|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>
| 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." [https://arxiv.org/abs/2107.12475 arxiv preprint] [https://www.erdosproblems.com/406 Erdős Problems]
|-
| BB(14)
| BBi(27)
| <div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}</div></div></div>
|Oct 2026
|[[User:Cosmo|Tristan Stérin]]
|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." Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
| Erdős Ternary Conjecture Variant
| BB(4,4)
| BBi(15)
| {{TM|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|Oct 2026
|roromachine14
|The machine halts if and only if the following conjecture is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1." Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
| Sloane Ternary Conjecture
| BB(6,4)
| BBi(21)
|style="width:30%;word-break:break-all"| {{TM|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if the following conjecture by N. J. A. Sloane is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n." Not independently verified. [http://neilsloane.com/doc/persistence.html Sloane's Page]
|-
| rowspan="2"|Weak Collatz
|BB(124) <br> 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.
|-
|BB(14) <br> BB(10,3)
|BBi(26)
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz (includes Lean formalization)
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}<br>{{TM|2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ|undecided}}</div></div></div>
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if the "weak Collatz conjecture" is false, as above. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|-
| rowspan="2"|Odd Perfect
|BB(23,3)
|BBi(63)
|<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}</div></div></div>
|Jul 2024
|kg583
|The machine halts if and only if there exists an odd multiply perfect number greater than 1.
Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
|BB(37)
|BBi(71)
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if there exists an odd perfect number. Not independently verified.
|-
|Sierpiński
|BB(60)
|BBi(104)
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if 78557 is the smallest Sierpiński number. Not independently verified.
|-
|Riesel
|BB(68)
|BBi(112)
|style="width:30%;word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel (includes Lean formalization)
|Oct 2026
|[[User:Loader3229]]
|The machine halts if and only if 509203 is the smallest Riesel number. Not independently verified.
|-
| rowspan="2"|Fermat's Last Theorem
|BB(400)
|
|''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965), pp. 81--82
|1965
|James Randels
|From Randels' doctoral dissertation ''[http://rave.ohiolink.edu/etdc/view?acc_num=osu1486567232687544 Studies in Turing Machines]'' (1965). 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.
|-
|BB(90)
|BBi(160)
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat (includes Lean formalization)
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine halts if and only if Fermat's Last Theorem is false, so it never halts. Not independently verified.
|-
| Bigfoot - compiled
| [[BB(7)]]
| BBi(13)
|style="width:30%;word-break:break-word"| {{TM|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| June 2024
| <code>@Iijil1</code>
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
| Hydra - compiled
|BB(9)
|BBi(15)
|style="width:30%;word-break:break-word"|{{TM|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}

[[File:Hydra_9_states.txt]]
|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/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. 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. [[Mother of Giants]] are all TMs formed by filling in the missing transition in <code>1RB1LE_0LC0LB_0LD1LC_1RD1RA_---0LA</code> with different values.

For a math problem encoded into a Π<sub>2</sub> statement, it can't encode into a TM halting problem, but it can encode into a TM quasihalting problem, so a BBB cryptid can be constructed from this problem.

{| class="wikitable"
|-
! Name !! BBB domain !! Machine !! Date !! Discoverer !! Note
|-
|[[Mother of Giants]]
|[[Beeping Busy Beaver|BBB(5)]]
|{{TM|1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA|non_halt}} and 3 others
|Apr 2022
|[[User:Sligocki|Shawn Ligocki]]
|Probviously quasihalting. Found "in the wild".
|-
|Twin Prime
|[[Beeping Busy Beaver|BBB(18)]]
|style="width:30%;word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/twinprimes
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM|non_halt}}</div><br>Beeping State: N</div></div>
|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|The machine [[quasihalt]]s if and only if Twin Prime conjecture is false. Not independently verified.
|-
|Collatz
|[[Beeping Busy Beaver|BBB(17)]]
|style="width:30%;word-break:break-word"|https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
<div class="toccolours mw-collapsible mw-collapsed">Machine code:<div class="mw-collapsible-content"><div style="word-break:break-all">{{TM|1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ|non_halt}}</div><br>Beeping State: O</div></div>
|Oct 2026
|[https://github.com/Mira-Cult Mira Cult]
|The machine [[quasihalt]]s if and only if the full [https://en.wikipedia.org/wiki/Collatz_conjecture Collatz conjecture] is false. Not independently verified.
|}

== Cryptids for other BB-style functions ==

{| class="wikitable"
|-
! Name !! Domain !! Machine !! Date !! Discoverer !! Note
|-
|
|Size 23 BitChanger programs
|<code><nowiki>}}}<[}<}[}<<<]}}<[<<]}}]*</nowiki></code>
|Apr 2025
|mxdys
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
|-
|[[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
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Frankenstein's Monster|Frankenstein's Monster]]
|[[Fractran|BBf(23)]]
|<code>[1/12, 9/10, 14/3, 121/2, 5/7, 3/11]</code>
|Dec 2025
|[[User:Sligocki|Shawn Ligocki]]
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Antihydra-like Cryptid|Antihydra-like Cryptid]]
|[[Fractran|BBf(23)]]
|<code>[9/10, 1/6, 1331/2, 14/3, 5/7, 3/11]</code>
|Dec 2025
|maksandchael
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|[[Fractran#Size 23: 11 Hydra-like Cryptids|11 Hydra-like Cryptids]]
|[[Fractran|BBf(23)]]
|[[Fractran#Size 23: 11 Hydra-like Cryptids|too many to display - see page]]
|Aug 2026
|Jason Yuen (@-d) and Claude Opus 4.6
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
|-
|[[Fractran#Size 25: Hydra|Hydra - compiled]]
|[[Fractran|BBf(25)]]
|<code>[363/14, 125/2, 22/21, 1/3, 7/11, 14/5]</code>
|Dec 2025
|maksandchael
|Compilation of Hydra.
|-
|[[Fractran#Size 36: BMO1|BMO 1 - compiled]]
|[[Fractran|BBf(36)]]
|<code>[153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3]</code>
|Nov 2025
|maksandchael
|Compilation of BMO 1.
|-
|
|[[Fractran|BBf(48)]]
|<code>[77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13]</code>
|Nov 2025
|maksandchael
|Similar to Space Needle, probviously non-halting.
|-
|
|[[Lambda Calculus|BBλ(74)]]
|<math>(\lambda1111(\lambda1)1(\lambda\lambda1))(\lambda\lambda\lambda1(322222))</math>
|Apr 2025
|50_ft_lock
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[CounterScript|BBCS(23)]]
|<code><nowiki>A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;}</nowiki></code>
|Jun 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Post Tag System|BB_PT(11)]]
|<code>11111_20_1</code>
|Jul 2026
|sheep, [[User:Sligocki|Shawn Ligocki]]
|This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
|-
|
|[[Pebble Automaton|peBBle(12)]]
|<code>00_00_00_40_10_00_00_08_41_11_00_a1</code>
|Oct 2026
|[[User:AlephSquirrel]]
|Similar to BMO 1.
|-
|
|[[Pebble Automaton|peBBle(86)]]
|https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569
|Sep 2026
|sheep
|Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
|-
|
|[[Register machine|MBB(8)]]
|<code>0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B*</code>
|Oct 2026
|[[User:Sligocki|Shawn Ligocki]]
|Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
|-
|[[Register machine#MBB(8): Space Needle variants|Needle 1]]
|[[Register machine|MBB(8)]]
|<code>0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C*</code>
|Oct 2026
|sheep
|Similar to Space Needle, probviously non-halting. Found "in the wild".
|-
|[[Register machine#MBB(10): Hydra|Hydra - compiled]]
|[[Register machine|MBB(10)]]
|<code>0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA</code>
|Jun 2025
|[[User:Sligocki|Shawn Ligocki]]
|Compilation of Hydra.
|-
|[[Register machine#MBB(12,2): Convergence of 7 under the 5n + 1 map|5n+1]]
|[[Register machine|MBB(12,2)]]
|<code>0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H*</code>
|Oct 2026
|sheep
|This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
|-
|Erdős Ternary Conjecture
|[[Register machine|MBB(19)]]
|<code>4+B_4+C_0+D_0+E_2-CF_0-GI_2+H_1+F_1-JN_1-KN_1-LM_3+I_0+N_3-OQ_1+P_3-OI_0-RS_0-RE_4-E*</code>
|Sep 2026
|mineralli
|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."
|-
|RH
|[[Register machine|MBB(129)]]
|https://www.sciencedirect.com/science/article/pii/S0304397519304633
|Feb 2020
|Yuri Matiyasevich
|The machine halts if and only if the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|-
|
|[[Boolfuck|BB_boolf(24)]]
|<code>*>*>*[>[*<<]*>*[<<]*>*>]</code>
|Oct 2026
|sheep
|Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
|-
|ZFC
|[[Wang Tiles|BB<sub>WT</sub>(11487)]]
|https://github.com/jconorgrogan/k230-wang-tiles-lean
|Oct 2026
|[https://github.com/jconorgrogan jconorgrogan]
|The machine halts if and only if [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent. Not independently verified.
|}

== References ==
[[Category:Zoology]]
[[Category:Cryptids]]

Cryptids

2026-10-25T12:20:44Z

Frogmatic: /* Larger Cryptids */ Remove redundant BB entries. There's no great value in highlighting a BB(n) machine when a BB(<n) machine is mentioned in the same row.

Cryptids

2026-10-25T12:18:39Z

Frogmatic: /* Larger Cryptids */ Break instruction count out into a separate column (where provided). This helps to keep the table more readable I think.

Cryptids

2026-10-24T17:18:00Z

Frogmatic: /* Larger Cryptids */ Add the full Collatz machine

Cryptids

2026-10-23T18:09:12Z

Frogmatic: /* Larger Cryptids */ Move the "Announcement" concept into general notes, since most rows don't have a formal announcement now.

1LM0RB 1LC1RB 0LM0LD 1RE1LD 0RF0RB 0LN0LG 1LH1LG 0RK0RI 1RJ1RI 0LN1RF 0RL1RK 0RZ1LC 0RA1LM 1LM1LN

2026-10-22T21:22:51Z

Frogmatic: Fix MathJAX syntax and link my user page

{{machine|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN}}
{{TM|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN}} is a 14-state, 2-symbol [[TM]] that halts iff the weak Collatz conjecture is false. It was found by [[User:Frogmatic|Charlie Reams]] with the help of @.mira.mira. using AI tools (see https://github.com/calr3/tinycryptids/blob/weakcollatz-14/weakcollatz/README.md).

== Weak Collatz ==
Let <math display=block>
T(n) = \begin{cases}
\frac{n}{2} & \text{if } n \equiv 0 \pmod{2} \\
\frac{3n+1}{2} & \text{if } n \equiv 1 \pmod{2} \\
\end{cases}
</math>
be the (shortcut) Collatz function.

The weak Collatz conjecture is that the only cycle (for positive integers) of the iterated Collatz function is the trivial 1 -> 2 -> 1 cycle. We can state this formally as <math>\forall M \geq 3, \; \forall k \geq 1, \; T^k(M) \neq M</math> which is a <math>\Pi_1</math> statement and thus encodable into a TM halting problem. In contrast, there is no known encoding of the full Collatz conjecture into a <math>\Pi_1</math> statement. The typical formal statement <math>\forall M \ge 1, \exists k, T^k(M) = 1</math> is a <math>\Pi_2</math> statement.

This TM depends on an even simpler algorithm: for each M ≥ 3, start with v = M, repeatedly apply v = T(v) until:
* v = 1 or v > M: Move on to next M
* v = M: Halt

For any non-trivial cycle, this will eventually be detected by this algorithm when M reaches the maximum value in that cycle.

Note that this inner loop (testing <math>T^k(M)</math>) must complete within M steps. By the Pigeonhole principle, after M steps if non were 1 nor ≥ M then there must be at least one repeated value <math>x \in [2, M-1]</math>. That is a cycle. But we would have noticed that cycle earlier when simulating M=x (or, more correctly, M = the maximum value in the cycle of x).

Therefore this algorithm explores all M values until either it finds the max value of a non-trivial cycle and halts or runs forever if no such cycle exists.

== Encoding ==
This TM tracks it's current (v,M) values in config: <code>C(v, M) = 0^inf A> 1^v-1 0 1^M-v 0^inf</code> (in other words, a string of M 1s with the v-th index set to 0) and it follows the rules:

<pre>
C(1, M) -> C(M+1, M+1)
C(2k, M) -> C(k, M)
if M > 3k+2: C(2k+1, M) -> C(3k+2, M)
if M = 3k+2: C(2k+1, M) -> Halt
if M < 3k+2: C(2k+1, M) -> C(M+1, M+1)
</pre>

Cryptids

2026-10-22T20:53:12Z

Frogmatic: /* Larger Cryptids */ Better table formatting for problems with multiple machines

Cryptids

2026-10-22T18:52:49Z

Frogmatic: /* Larger Cryptids */ Note verification of weak Collatz in BB(14).

Cryptids

2026-10-22T18:47:36Z

Frogmatic: Give DeepMind the credit for Goldback BB(21), since it's from their repo.

User:Frogmatic

2026-10-18T12:22:59Z

Frogmatic: Page for myself, red links begone

Charlie Reams, being a human person and theoretical CS dabbler. [https://www.cl.cam.ac.uk/techreports/UCAM-CL-TR-821.html Cambridge PhD], now at Google.

Cryptids

2026-10-18T12:19:33Z

Frogmatic: /* Larger Cryptids */ Mention possible improvement to the Weak Collatz cryptid