BusyBeaverWiki - User contributions [en]

Cryptids

2026-11-09T07:41:17Z

Veljjanoski: Brocard row: machine reduced to 34 states (BB(34), BBi(66)); links updated

[[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="3"| '''[[Logical independence|ZFC consistency]]'''
|rowspan="3"| 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 Elliot Glazer].
|-
|'''BB(152,9)'''
'''BB(143,11)'''
'''BB(117,12)'''
|'''BBi(376)'''
|
|[https://github.com/calr3/tinycryptids/tree/main/zfc_nsymbols calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Nov 2026
|[[User:Frogmatic|Charlie Reams]]
|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="3"|'''Goldbach'''
| rowspan="3"|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.
|-
|'''BB(10,6)'''
|'''BBi(34)'''
|{{WrapTM|Machine code|1RD3LB---------5RF_0RD2RC2LB3LB---5RD_0LE2LB2RC3RC2LB5LE_1LD1LI1RD3LB------_0RH1RE1LE4LE1RA1LI_---1RG------------_---1RA------------_---1RH------1RH1LI_1RF5LJ------------_0RF1LJ---------1LJ|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/goldbach calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|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) '''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]. The BB(15) machine's behavior has been verified in Lean.<ref>https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/ErdosTernary15.lean</ref>
|-
| '''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]]
|Its behavior has been verified in Lean.<ref>https://github.com/google-deepmind/cryptids/tree/main/artifacts/erdos_ternary_14</ref>
|-
| '''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) 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)''' '''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.
|-
| '''Brocard'''
| Halts iff [[wikipedia:Brocard's problem|Brocard's problem]] <code>n! + 1 = m2</code> has a solution with <code>n > 7</code>, i.e. beyond the known solutions <code>n = 4, 5, 7</code>.
| '''BB(34)'''
| '''BBi(66)'''
|[https://github.com/veljjanoski/cryptids/blob/brocard-41/artifacts/brocard_34/machine.txt Machine code]
|[https://github.com/veljjanoski/cryptids/tree/brocard-41/artifacts/brocard_34 GitHub] ([https://github.com/google-deepmind/cryptids/pull/27 PR to google-deepmind/cryptids])
|style="white-space: nowrap; text-align: right;"|Nov 2026
|[https://www.linkedin.com/in/danielveljjanoski Daniel Veljjanoski]
|Keeps <code>n!/8</code> in unary and tests it for triangularity (<code>n! + 1 = m2</code> forces <code>m = 2k + 1</code>, <code>n!/8 = k(k+1)/2</code>). Includes a Lean formalization (kernel-checked; PR pending).<ref>https://github.com/veljjanoski/cryptids/blob/brocard-41/Cryptids/Semantics/Brocard34.lean</ref> An earlier 61-state machine with a Lean proof is by [[User:Frogmatic|Charlie Reams]] ([https://github.com/calr3/tinycryptids/tree/main/brocard calr3 GitHub], Nov 2026); a 43-state table without a machine-checked proof is in [https://arxiv.org/abs/2605.20215 arXiv:2605.20215].
|-
| '''Kurepa'''
| Halts if and only if Kurepa's conjecture is false.
| '''BB(38)'''
| '''BBi(65)'''
|[https://github.com/calr3/tinycryptids/blob/main/kurepa/machines/kurepa_38.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/kurepa 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.
[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].
|}

== 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 Π2 statement but not a Π1 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(\lambda2))(\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|BBWT(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

2026-11-09T06:37:42Z

Veljjanoski: Brocard row: link machine.txt instead of WrapTM (41 states exceed the 26-letter standard format; the bbchallenge viewer link rendered the machine wrongly)

[[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="3"| '''[[Logical independence|ZFC consistency]]'''
|rowspan="3"| 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 Elliot Glazer].
|-
|'''BB(152,9)'''
'''BB(143,11)'''
'''BB(117,12)'''
|'''BBi(376)'''
|
|[https://github.com/calr3/tinycryptids/tree/main/zfc_nsymbols calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Nov 2026
|[[User:Frogmatic|Charlie Reams]]
|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="3"|'''Goldbach'''
| rowspan="3"|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.
|-
|'''BB(10,6)'''
|'''BBi(34)'''
|{{WrapTM|Machine code|1RD3LB---------5RF_0RD2RC2LB3LB---5RD_0LE2LB2RC3RC2LB5LE_1LD1LI1RD3LB------_0RH1RE1LE4LE1RA1LI_---1RG------------_---1RA------------_---1RH------1RH1LI_1RF5LJ------------_0RF1LJ---------1LJ|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/goldbach calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|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) '''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]. The BB(15) machine's behavior has been verified in Lean.<ref>https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/ErdosTernary15.lean</ref>
|-
| '''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]]
|Its behavior has been verified in Lean.<ref>https://github.com/google-deepmind/cryptids/tree/main/artifacts/erdos_ternary_14</ref>
|-
| '''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) 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)''' '''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.
|-
| '''Brocard'''
| Halts iff [[wikipedia:Brocard's problem|Brocard's problem]] <code>n! + 1 = m2</code> has a solution with <code>n > 7</code>, i.e. beyond the known solutions <code>n = 4, 5, 7</code>.
| '''BB(41)'''
| '''BBi(72)'''
|[https://github.com/veljjanoski/cryptids/blob/brocard-41/artifacts/brocard_41/machine.txt Machine code]
|[https://github.com/veljjanoski/cryptids/tree/brocard-41/artifacts/brocard_41 GitHub] ([https://github.com/google-deepmind/cryptids/pull/27 PR to google-deepmind/cryptids])
|style="white-space: nowrap; text-align: right;"|Nov 2026
|[https://www.linkedin.com/in/danielveljjanoski Daniel Veljjanoski]
|Keeps <code>n!/8</code> in unary and tests it for triangularity (<code>n! + 1 = m2</code> forces <code>m = 2k + 1</code>, <code>n!/8 = k(k+1)/2</code>). Includes a Lean formalization (kernel-checked; PR pending).<ref>https://github.com/veljjanoski/cryptids/blob/brocard-41/Cryptids/Semantics/Brocard41.lean</ref> An earlier 61-state machine with a Lean proof is by [[User:Frogmatic|Charlie Reams]] ([https://github.com/calr3/tinycryptids/tree/main/brocard calr3 GitHub], Nov 2026); a 43-state table without a machine-checked proof is in [https://arxiv.org/abs/2605.20215 arXiv:2605.20215].
|-
| '''Kurepa'''
| Halts if and only if Kurepa's conjecture is false.
| '''BB(38)'''
| '''BBi(65)'''
|[https://github.com/calr3/tinycryptids/blob/main/kurepa/machines/kurepa_38.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/kurepa 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.
[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].
|}

== 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 Π2 statement but not a Π1 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(\lambda2))(\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|BBWT(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

2026-11-07T08:53:14Z

Veljjanoski:

[[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="3"| '''[[Logical independence|ZFC consistency]]'''
|rowspan="3"| 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 Elliot Glazer].
|-
|'''BB(152,9)'''
'''BB(143,11)'''
'''BB(117,12)'''
|'''BBi(376)'''
|
|[https://github.com/calr3/tinycryptids/tree/main/zfc_nsymbols calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Nov 2026
|[[User:Frogmatic|Charlie Reams]]
|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="3"|'''Goldbach'''
| rowspan="3"|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.
|-
|'''BB(10,6)'''
|'''BBi(34)'''
|{{WrapTM|Machine code|1RD3LB---------5RF_0RD2RC2LB3LB---5RD_0LE2LB2RC3RC2LB5LE_1LD1LI1RD3LB------_0RH1RE1LE4LE1RA1LI_---1RG------------_---1RA------------_---1RH------1RH1LI_1RF5LJ------------_0RF1LJ---------1LJ|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/goldbach calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|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) '''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]. The BB(15) machine's behavior has been verified in Lean.<ref>https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/ErdosTernary15.lean</ref>
|-
| '''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]]
|Its behavior has been verified in Lean.<ref>https://github.com/google-deepmind/cryptids/tree/main/artifacts/erdos_ternary_14</ref>
|-
| '''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) 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)''' '''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.
|-
| '''Brocard'''
| Halts iff [[wikipedia:Brocard's problem|Brocard's problem]] <code>n! + 1 = m2</code> has a solution with <code>n > 7</code>, i.e. beyond the known solutions <code>n = 4, 5, 7</code>.
| '''BB(41)'''
| '''BBi(72)'''
|{{WrapTM|Machine code|1LB---_1LC---_1LD---_0LE---_1LF---_1LG---_1LH---_1Rn---_0LJ1LI_1RK1LJ_0RR0RL_0RQ1RL_0RN1RM_1LO1RN_0LP1LO_1RQ1LP_0LI0RM_1RS1RR_0RT1LS_---0RU_0RV1RU_1LW1RV_0LX0LY_0Le1RU_0LZ1LY_1Ra1LZ_0Rb0RU_0Rc0Rb_0Ld0Rc_0Ld0LI_0Lf1Le_0Rn1Lg_0Rn1Lh_0Rn1Li_0Rn1Lj_0Rn1Lk_0Rn1Ll_0Rn1Lm_0Rn---_0Ro1Rn_1Rb1Ro|undecided}}
|[https://github.com/veljjanoski/cryptids/tree/brocard-41/artifacts/brocard_41 GitHub] ([https://github.com/google-deepmind/cryptids/pull/27 PR to google-deepmind/cryptids])
|style="white-space: nowrap; text-align: right;"|Nov 2026
|[https://www.linkedin.com/in/danielveljjanoski Daniel Veljjanoski]
|Keeps <code>n!/8</code> in unary and tests it for triangularity (<code>n! + 1 = m2</code> forces <code>m = 2k + 1</code>, <code>n!/8 = k(k+1)/2</code>). Includes a Lean formalization (kernel-checked; PR pending).<ref>https://github.com/veljjanoski/cryptids/blob/brocard-41/Cryptids/Semantics/Brocard41.lean</ref> An earlier 61-state machine with a Lean proof is by [[User:Frogmatic|Charlie Reams]] ([https://github.com/calr3/tinycryptids/tree/main/brocard calr3 GitHub], Nov 2026); a 43-state table without a machine-checked proof is in [https://arxiv.org/abs/2605.20215 arXiv:2605.20215].
|-
| '''Kurepa'''
| Halts if and only if Kurepa's conjecture is false.
| '''BB(38)'''
| '''BBi(65)'''
|[https://github.com/calr3/tinycryptids/blob/main/kurepa/machines/kurepa_38.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/kurepa 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.
[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].
|}

== 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 Π2 statement but not a Π1 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(\lambda2))(\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|BBWT(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

2026-11-06T18:47:45Z

Veljjanoski: →Larger Cryptids: Erdős Ternary BB(15) and BB(14) machines verified in Lean (google-deepmind/cryptids PR 19)

[[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="3"| '''[[Logical independence|ZFC consistency]]'''
|rowspan="3"| 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 Elliot Glazer].
|-
|'''BB(152,9)'''
'''BB(143,11)'''
'''BB(117,12)'''
|'''BBi(376)'''
|
|[https://github.com/calr3/tinycryptids/tree/main/zfc_nsymbols calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Nov 2026
|[[User:Frogmatic|Charlie Reams]]
|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="3"|'''Goldbach'''
| rowspan="3"|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.
|-
|'''BB(10,6)'''
|'''BBi(34)'''
|{{WrapTM|Machine code|1RD3LB---------5RF_0RD2RC2LB3LB---5RD_0LE2LB2RC3RC2LB5LE_1LD1LI1RD3LB------_0RH1RE1LE4LE1RA1LI_---1RG------------_---1RA------------_---1RH------1RH1LI_1RF5LJ------------_0RF1LJ---------1LJ|undecided}}
|[https://github.com/calr3/tinycryptids/tree/main/goldbach calr3 GitHub]
|style="white-space: nowrap; text-align: right;"|Oct 2026
|[[User:Frogmatic|Charlie Reams]]
|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) '''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]. The BB(15) machine's behavior has been verified in Lean.<ref>https://github.com/google-deepmind/cryptids/blob/main/Cryptids/Semantics/ErdosTernary15.lean</ref>
|-
| '''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]]
|Its behavior has been verified in Lean.<ref>https://github.com/google-deepmind/cryptids/tree/main/artifacts/erdos_ternary_14</ref>
|-
| '''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) 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)''' '''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.
|-
| '''Kurepa'''
| Halts if and only if Kurepa's conjecture is false.
| '''BB(38)'''
| '''BBi(65)'''
|[https://github.com/calr3/tinycryptids/blob/main/kurepa/machines/kurepa_38.txt Machine code]
|[https://github.com/calr3/tinycryptids/tree/main/kurepa 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.
[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].
|}

== 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 Π2 statement but not a Π1 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(\lambda2))(\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|BBWT(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]]