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== Larger Cryptids ==
== 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".
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"
{| class="wikitable"
|-
|-
! Name !! Behaviour !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
! Name !! Behaviour !! [[Busy Beaver Functions|BB Domain]] !! [[Instruction-Limited Busy Beaver|Instructions]] !! Machine !! Date !! Discoverer !! Note
|-
|-
|[[Logical independence|ZF consistency]]
|'''[[Logical independence|ZF consistency]]'''
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|BB(432)
|'''BB(432)'''
|
|
|style="word-break:break-word"|[https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e Wade's machine]
|style="word-break:break-word"|[https://codeberg.org/ajwade/turing_machine_explorer/src/commit/33b30300054242201a95679aacf7e74312bd8803b0df9a85d2314095efd6804e Wade's machine]
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|
|
|-
|-
|ZFC consistency
|'''ZFC consistency'''
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|Halts iff [[wikipedia:Zermelo–Fraenkel_set_theory|Zermelo–Fraenkel set theory]] is inconsistent.
|BB(215)
|'''BB(215)'''
|
|
|style="word-break:break-word"|https://github.com/elliotglazer/k215-pa
|style="word-break:break-word"|https://github.com/elliotglazer/k215-pa
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|Machine-checked in Peano-Arithmetic. Not independently verified.
|Machine-checked in Peano-Arithmetic. Not independently verified.
|-
|-
|Peano Arithmetic consistency
|'''Peano Arithmetic consistency'''
|Halts iff Peano Arithmetic is inconsistent
|Halts iff Peano Arithmetic is inconsistent
|BB(372)
|'''BB(372)'''
|BBi(742)
|'''BBi(742)'''
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py
|style="white-space: nowrap; text-align: right;"|2026
|style="white-space: nowrap; text-align: right;"|2026
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|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 Discord announcement]
|-
|-
| rowspan="3" | Riemann Hypothesis
| rowspan="3" | '''Riemann Hypothesis'''
| rowspan="3" | Halts iff the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
| rowspan="3" | Halts iff the [[wikipedia:Riemann_hypothesis|Riemann hypothesis]] is false.
|BB(744)
|BB(744)
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|
|
|-
|-
|BB(95)
|'''BB(95)'''
|BBi(189)
|'''BBi(189)'''
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95
|style="white-space: nowrap; text-align: right;"|Oct 2026
|style="white-space: nowrap; text-align: right;"|Oct 2026
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|Not independently verified.
|Not independently verified.
|-
|-
| rowspan="2"|Goldbach
| rowspan="2"|'''Goldbach'''
| rowspan="2"|Halts iff [[wikipedia:Goldbach's conjecture|Goldbach's conjecture]] is false
| rowspan="2"|Halts iff [[wikipedia:Goldbach's conjecture|Goldbach's conjecture]] is false
|BB(25)
|BB(25)
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|Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|Its behavior has been verified in Lean.<ref>https://github.com/lengyijun/goldbach_tm</ref>
|-
|-
|BB(21)
|'''BB(21)'''
|BBi(41)
|'''BBi(41)'''
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
|style="word-break:break-word"|https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
{{WrapTM|Machine code|1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU|undecided}}
{{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}}
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|Not independently verified.
|Not independently verified.
|-  
|-  
| rowspan="2"| Erdős Ternary Conjecture
| rowspan="2"| '''Erdős Ternary Conjecture'''
| rowspan="2"| Halts iff [https://www.erdosproblems.com/406 Erdős Problem 406] is false.
| rowspan="2"| Halts iff [https://www.erdosproblems.com/406 Erdős Problem 406] is false.
| BB(15) <br> BB(5,4)
| BB(15) <br> '''BB(5,4)'''
| BBi(19)
| '''BBi(19)'''
|style="word-break:break-word"|
|style="word-break:break-word"|
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
https://docs.bbchallenge.org/other/powers_of_two_15_2.txt
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| 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]
| Erdős #406 is the conjecture that "For all n > 8, there is at least one 2 in the base-3 representation of 2^n." [https://arxiv.org/abs/2107.12475 arxiv preprint]
|-
|-
| BB(14)
| '''BB(14)'''
| BBi(27)
| BBi(27)
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
| {{WrapTM|Machine code|1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM|undecided}}
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|Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|Not independently verified. [https://discord.com/channels/960643023006490684/1561295445312151622/1562203343114473472 Announced on Discord]
|-
|-
| Erdős Ternary Conjecture Variant
| '''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."
| 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)
| '''BB(4,4)'''
| BBi(15)
| '''BBi(15)'''
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
| {{WrapTM|Machine code|1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|style="white-space: nowrap; text-align: right;"|Oct 2026
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| Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
| Not independently verified. [https://discord.com/channels/960643023006490684/1084047886494470185/1563042146649382983 Announced on Discord]
|-
|-
| Sloane Ternary Conjecture
| '''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."
| 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)
| '''BB(6,4)'''
| BBi(21)
| '''BBi(21)'''
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
| {{WrapTM|Machine code|2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|style="white-space: nowrap; text-align: right;"|Oct 2026
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| Not independently verified.  
| Not independently verified.  
|-
|-
| rowspan="2"|Weak Collatz
| rowspan="2"| '''Weak Collatz'''
| rowspan="2"|Halts iff the weak Collatz conjecture is false.  
| rowspan="2"|Halts iff the weak Collatz conjecture is false.  
|BB(124) <br> BB(43,4)
|BB(124) <br> '''BB(43,4)'''
|BBi(121)
|BBi(121)
|style="word-break:break-word"|https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt
|style="word-break:break-word"|https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt
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|The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). Not independently verified, and probably easy to further optimise.
|The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). Not independently verified, and probably easy to further optimise.
|-
|-
|BB(14) <br> BB(10,3)
|'''BB(14)''' <br> '''BB(10,3)'''
|BBi(26)
|'''BBi(26)'''
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/weakcollatz
{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
{{WrapTM|BB(14) machine code|1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN|undecided}}
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|Includes Lean formalization. [https://github.com/sligocki/busy-beaver/blob/main/rust/src/validator.rs#L3900 BB(14) verified] by Shawn Ligocki.
|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
| '''Odd multiply perfect'''
| Halts iff an odd [[wikipedia:Multiply perfect number|multiply perfect]] greater than 1 exists.
| Halts iff an odd [[wikipedia:Multiply perfect number|multiply perfect]] greater than 1 exists.
|BB(23,3)
| '''BB(23,3)'''
|BBi(63)
| '''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}}
|{{WrapTM|Machine code|1RO0RB2RF_1LL1RB2RC_0RC0LD2LI_0LD1LD2LE_1RA1LE0RA_0RF1LG2LM_0LG1LG2LH_0RA1LH0RA_1LI---2LJ_1LK1LJ1LK_1LL1LK---_2RA1LS---_1LM---2LN_0RO1LN2RR_2LW0RP2LJ_1LQ1RP2RP_1RO1LQ2LQ_2LG1RR2RR_0RT0LS0LS_0RT1RT2RU_0LV0RU0RU_0LV---2LW_2LB1LW---|undecided}}
|style="white-space: nowrap; text-align: right;"|Jul 2024
|style="white-space: nowrap; text-align: right;"|Jul 2024
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|Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|Not independently verified. [https://discord.com/channels/960643023006490684/1026577255754903572/1259918765437747239 Discord]
|-
|-
| Odd perfect
| '''Odd perfect'''
| Halts iff an odd [[wikipedia:Perfect number|perfect]] exists.
| Halts iff an odd [[wikipedia:Perfect number|perfect]] exists.
|BB(37)
| '''BB(37)'''
|BBi(71)
| '''BBi(71)'''
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/oddperfect
|style="white-space: nowrap; text-align: right;"|Oct 2026
|style="white-space: nowrap; text-align: right;"|Oct 2026
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|Includes Lean formalization. Not independently verified.
|Includes Lean formalization. Not independently verified.
|-
|-
|Sierpiński  
| '''Sierpiński'''
|Halts iff 78,557 is the smallest [[wikipedia:Sierpiński number|Sierpiński number]].  
|Halts iff 78,557 is the smallest [[wikipedia:Sierpiński number|Sierpiński number]].  
|BB(60)
| '''BB(60)'''
|BBi(104)
| '''BBi(104)'''
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/sierpinski
|style="white-space: nowrap; text-align: right;"|Oct 2026
|style="white-space: nowrap; text-align: right;"|Oct 2026
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|Includes Lean formalization. Not independently verified.  
|Includes Lean formalization. Not independently verified.  
|-
|-
|Riesel
| '''Riesel'''
|Halts iff 509,203 is the smallest [[wikipedia:Riesel number|Riesel number]].  
|Halts iff 509,203 is the smallest [[wikipedia:Riesel number|Riesel number]].  
|BB(68)
| '''BB(68)'''
|BBi(112)
| '''BBi(112)'''
|style="word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel
|style="word-break:break-word"|https://github.com/loader3229/tinycryptids/tree/main/riesel
|style="white-space: nowrap; text-align: right;"|Oct 2026
|style="white-space: nowrap; text-align: right;"|Oct 2026
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|Includes Lean formalization. Not independently verified.  
|Includes Lean formalization. Not independently verified.  
|-
|-
| rowspan="2"|Fermat's Last Theorem
| rowspan="2"| '''Fermat's Last Theorem'''
| rowspan="2"|Does not halt.
| rowspan="2"|Does not halt.
|BB(400)
|BB(400)
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|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.
|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)
| '''BB(90)'''
|BBi(160)
| '''BBi(160)'''
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat
|style="word-break:break-word"|https://github.com/calr3/tinycryptids/tree/main/fermat
|style="white-space: nowrap; text-align: right;"|Oct 2026
|style="white-space: nowrap; text-align: right;"|Oct 2026
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|Includes Lean formalization. Not independently verified.
|Includes Lean formalization. Not independently verified.
|-
|-
| Bigfoot - compiled
| '''Bigfoot''' - compiled
| Halts iff [[Bigfoot]] halts.
| Halts iff [[Bigfoot]] halts.
| [[BB(7)]]
| '''[[BB(7)]]'''
| BBi(13)  
| '''BBi(13)'''
| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
| {{WrapTM|Machine code|0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB|undecided}}
|style="white-space: nowrap; text-align: right;"| June 2024
|style="white-space: nowrap; text-align: right;"| June 2024
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| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
| Compilation of Bigfoot into 2 symbols, there was a previous [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-1774200442 compilation with 8 states]. [https://github.com/sligocki/sligocki.github.io/issues/8#issuecomment-2140887228 Bigfoot Comment]
|-
|-
| Hydra - compiled
| '''Hydra''' - compiled
| Halts iff [[Hydra]] halts.
| Halts iff [[Hydra]] halts.
|BB(9)
| '''BB(9)'''
|BBi(15)
| '''BBi(15)'''
|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}
|{{WrapTM|Machine code|0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ|undecided}}
[[File:Hydra_9_states.txt]]
[[File:Hydra_9_states.txt]]
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<small>[https://discord.com/channels/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
<small>[https://discord.com/channels/960643023006490684/1084047886494470185/1251572501578780782 Announced on Discord]. Previous compilation had 10 states, by Daniel Yuan, also [https://discord.com/channels/960643023006490684/1084047886494470185/1247579473042346136 confirmed by Shawn Ligocki].</small>
|-
|-
| Antihydra-like
| '''Antihydra-like'''
|
|
|[[BB(7)]]
| '''[[BB(7)]]'''
|BBi(13)
| '''BBi(13)'''
|{{WrapTM|Machine code|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|{{WrapTM|Machine code|1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE|undecided}}
|style="white-space: nowrap; text-align: right;"|Oct 2026
|style="white-space: nowrap; text-align: right;"|Oct 2026

Revision as of 15:17, 25 October 2026

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.
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 blog post announcing the discovery of Bigfoot.

Cryptids at the Edge

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

Name BB/BBi domain Machine Date Discoverer Note
Bigfoot BB(3,3) BBi(8) 1RB2RA1LC_2LC1RB2RB_---2LA1LA (bbch) Nov 2023 Shawn Ligocki Probviously non-halting.
Hydra BB(2,5) BBi(9) 1RB3RB---3LA1RA_2LA3RA4LB0LB0LA (bbch) May 2024 Daniel Yuan Probviously non-halting.
Bonus cryptid BB(2,5) BBi(9) 1RB3RB---3LA1RA_2LA3RA4LB0LB1LB (bbch) May 2024 Daniel Yuan Probviously non-halting.
Antihydra BB(6) 1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA (bbch) Jun 2024 mxdys, shown to be a Cryptid by Racheline. Same as Hydra but starting iteration from 8 instead of 3 and with termination condition O > 2E instead of E > 2O, hence the name Antihydra.
Lucy's Moonlight BB(6) 1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA (bbch) Mar 2025 Racheline Probviously halting.
BB(6) 1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA (bbch) Jul 2024 mxdys Variant of Hydra and Antihydra. Probviously non-halting.
BB(6) 1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC (bbch) Aug 2024 mxdys Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
BB(6) 1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB--- (bbch) Sep 2024 Daniel Yuan Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
BB(6) 1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB (bbch) Nov 2024 Racheline Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
Space Needle BB(6) 1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD (bbch) Jan 2025 mxdys Probviously non-halting.
BB(6) 1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC (bbch) Jul 2024 mxdys Has near-identical behavior to 16 related BB(6) holdouts. Probviously halting.
BB(6) 1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF (bbch) Jul 2024 Racheline Probviously halting.
BB(6) 1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB (bbch) Jul 2024 Racheline Probviously halting.
BB(6) 1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA (bbch) Jul 2024 Racheline Probviously halting.
BB(6) 1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE (bbch) Nov 2024 Racheline Probviously halting.
BB(6) 1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA (bbch) Nov 2024 Racheline Probviously halting.
BB(6) 1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE (bbch) Jul 2025 mxdys Probviously halting.
BB(6) 1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE (bbch) 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) 1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC (bbch) Aug 2024 mxdys, shown to be a Cryptid by DrDisentangle Similar to Space Needle, probviously non-halting.
BB(6) 1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE (bbch) Apr 2026 sheep, shown to be a Cryptid by Daniel Yuan Similar to Space Needle, probviously non-halting.
BB(6) 1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB (bbch) 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.

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.

Name Behaviour BB Domain Instructions Machine Date Discoverer Note
ZF consistency Halts iff Zermelo–Fraenkel set theory is inconsistent. BB(432) Wade's machine 2025 Wade, based on work by CatIsFluffy and O'Rear
ZFC consistency Halts iff Zermelo–Fraenkel set theory is inconsistent. BB(215) https://github.com/elliotglazer/k215-pa Oct 2026 elliotglazer Machine-checked in Peano-Arithmetic. Not independently verified.
Peano Arithmetic consistency Halts iff Peano Arithmetic is inconsistent BB(372) BBi(742) https://github.com/LegionMammal978/turing_machine_explorer/blob/main/pa.py 2026 LegionMammal Discord announcement
Riemann Hypothesis Halts iff the Riemann hypothesis is false. BB(744) https://github.com/sorear/metamath-turing-machines/blob/master/machines/2016-riemann-matiyasevich-aaronson-744/riemann.nql 2016 Matiyasevich and O’Rear
BB(120) BBi(238) https://arxiv.org/abs/2609.30306 Sep 2026 Joseph M. Shunia
BB(95) BBi(189) https://github.com/google-deepmind/cryptids/tree/main/artifacts/riemann_95 Oct 2026 Google DeepMind Not independently verified.
Goldbach Halts iff Goldbach's conjecture is false BB(25) BBi(49) https://gist.github.com/anonymous/a64213f391339236c2fe31f8749a0df6
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 (bbch)
 May 2016 anonymous Its behavior has been verified in Lean.[1]
BB(21) BBi(41) https://github.com/google-deepmind/cryptids/tree/main/artifacts/goldbach_21
Machine code
1LJ0RF_0RJ1RD_1RP1LC_---1RE_0LO1RG_1LQ1RF_0RH0LH_0LM1LU_1RB0LI_1RJ0RK_1RL1RK_0LM0LS_1LN1LM_1LO0LC_1LQ1LO_1RD1RP_0RA0LR_0RB1LR_1LT1LS_0LU0LI_0RB0LU (bbch)
 Oct 2026 Google DeepMind Not independently verified.
Erdős Ternary Conjecture Halts iff Erdős Problem 406 is false. BB(15)
BB(5,4) 		BBi(19)

https://docs.bbchallenge.org/other/powers_of_two_15_2.txt

BB(15) machine
1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA (bbch)

https://docs.bbchallenge.org/other/powers_of_two_5_4.txt

BB(5,4) and BBi(19) machine
1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE (bbch)
 Jul 2021 Tristan Stérin (@cosmo) 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." arxiv preprint
BB(14) BBi(27)
Machine code
1LB1RA_1LC1LB_0RD0RA_1LM0RE_1LF1RE_0LJ1LG_1RK0LH_1RI1LH_0LJ0RE_0RD1LJ_0LL1RK_1RN0LC_0LN---_0RK0LM (bbch)
 Oct 2026 Tristan Stérin Not independently verified. Announced on Discord
Erdős Ternary Conjecture variant Halts iff the following conjecture (a variant of #406) is false: "For all n > 0, there is at least one 1 in the base-3 representation of 2^n - 1." BB(4,4) BBi(15)
Machine code
1RB2RA1RB3RA_0LC3RB2RB1RA_1RZ1LD2LC3LC_0RA1LD2LD3LD (bbch)
 Oct 2026 roromachine14 Not independently verified. Announced on Discord
Sloane Ternary Conjecture Halts if and only if the following conjecture by N. J. A. Sloane is false: "For all n > 15, there is at least one 0 in the base-3 representation of 2^n." BB(6,4) BBi(21)
Machine code
2RB2RB1RA3RA_0LC1RB3RB2RA_0LE1LD2LC3LC_0RB1LD2LD3LD_1RD2RD3RD0LF_1RE2RZ------ (bbch)
 Oct 2026 User:Loader3229 Not independently verified.
Weak Collatz Halts iff the weak Collatz conjecture is false. BB(124)
BB(43,4) 		BBi(121) https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt

https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt

Jul 2021 Tristan Stérin The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers (4 -> 2 -> 1 -> 4). Not independently verified, and probably easy to further optimise.
BB(14)
BB(10,3) 		BBi(26) https://github.com/calr3/tinycryptids/tree/main/weakcollatz
BB(14) machine code
1LM0RB_1LC1RB_0LM0LD_1RE1LD_0RF0RB_0LN0LG_1LH1LG_0RK0RI_1RJ1RI_0LN1RF_0RL1RK_0RZ1LC_0RA1LM_1LM1LN (bbch)
BB(10,3) and BBi(26) machine code
2RG0RB---_1LC1RB2LC_0LI0LD2LD_1RE1LD1LF_2RC0RB---_0RJ2RG---_1RH1RG1RH_2LI1RC1RH_0RA1LI1LI_0LI1RJ1RZ (bbch)
 Oct 2026 Charlie Reams Includes Lean formalization. BB(14) verified by Shawn Ligocki.
Odd multiply perfect Halts iff an odd multiply perfect greater than 1 exists. BB(23,3) BBi(63)
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--- (bbch)
 Jul 2024 kg583 Not independently verified. Discord
Odd perfect Halts iff an odd perfect exists. BB(37) BBi(71) https://github.com/calr3/tinycryptids/tree/main/oddperfect Oct 2026 Charlie Reams Includes Lean formalization. Not independently verified.
Sierpiński Halts iff 78,557 is the smallest Sierpiński number. BB(60) BBi(104) https://github.com/calr3/tinycryptids/tree/main/sierpinski Oct 2026 Charlie Reams Includes Lean formalization. Not independently verified.
Riesel Halts iff 509,203 is the smallest Riesel number. BB(68) BBi(112) https://github.com/loader3229/tinycryptids/tree/main/riesel Oct 2026 User:Loader3229 Includes Lean formalization. Not independently verified.
Fermat's Last Theorem Does not halt. BB(400) Studies in Turing Machines (1965), pp. 81--82 1965 James Randels From Randels' doctoral dissertation 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/tree/main/fermat Oct 2026 Charlie Reams Includes Lean formalization. Not independently verified.
Bigfoot - compiled Halts iff Bigfoot halts. BB(7) BBi(13)
Machine code
0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB (bbch)
 June 2024 @Iijil1 Compilation of Bigfoot into 2 symbols, there was a previous compilation with 8 states. Bigfoot Comment
Hydra - compiled Halts iff Hydra halts. BB(9) BBi(15)
Machine code
0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ (bbch)

File:Hydra 9 states.txt

June 2024 @Iijil1 Compilation of Hydra into 2 symbols, all confirmed by Shawn Ligocki. @Iijil1 provided 24 TMs which all emulate the same behavior.

Announced on Discord. Previous compilation had 10 states, by Daniel Yuan, also confirmed by Shawn Ligocki.

Antihydra-like BB(7) BBi(13)
Machine code
1LG0LD_1RE1RA_1RB1RC_1RC1LD_---0LF_1LC1LF_1LF1LE (bbch)
 Oct 2026 Jacobzheng 2n -> 3n, 2n+1 -> 3n+5, start at 2, halts only when the number of odd numbers is more than twice the number of even numbers.

Beeping Busy Beaver

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

For a math problem encoded into a Π2 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.

Name BBB domain Machine Date Discoverer Note
Mother of Giants BBB(5) 1RB1LE_0LC0LB_0LD1LC_1RD1RA_1RC0LA (bbch) and 3 others Apr 2022 Shawn Ligocki Probviously quasihalting. Found "in the wild".
Twin prime conjecture BBB(18) https://github.com/calr3/tinycryptids/tree/main/twinprimes
Machine code (beep state: N)
1LB1RA_0LH0LC_0LD1LC_1RE1LD_0LF0RG_0RE1LF_0RA1RG_1RP1LI_1RL1LJ_1RP1LK_1RL1LK_0RM1RL_---0RN_0LB1RO_0LB1RO_1LQ1RP_1RR1LQ_---0RM (bbch)
 Oct 2026 Charlie Reams The machine quasihalts if and only if Twin Prime conjecture is false. Not independently verified.
Collatz BBB(17) https://github.com/Mira-Cult/math/blob/main/turing-machines/collatz-beeping17/README.md
Machine code (beep state: O)
1RL0RB_1LC1RB_0LK0LD_1RE1LD_0RF0RB_0LG0LG_1LH1LG_0RA0RI_1RJ1RI_1RF1RF_0RA1LK_0LM0LM_0LN1LM_1RO1LN_0RA0RP_0RQ1RP_1RL1RQ (bbch)
 Oct 2026 Mira Cult The machine quasihalts if and only if the full Collatz conjecture is false. Not independently verified.

Cryptids for other BB-style functions

Name Domain Machine Date Discoverer Note
Size 23 BitChanger programs }}}<[}<}[}<<<]}}<[<<]}}]* Apr 2025 mxdys Similar random walk mechanism to Bigfoot. Probviously non-halting. Found "in the wild".
Fenrir BBf(22) [1/15, 27/77, 49/3, 10/49, 33/2] and 2 others Mar 2026 Jason Yuen (@-d) and Claude Opus 4.6 Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
Frankenstein's Monster BBf(23) [1/12, 9/10, 14/3, 121/2, 5/7, 3/11] Dec 2025 Shawn Ligocki Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
Antihydra-like Cryptid BBf(23) [9/10, 1/6, 1331/2, 14/3, 5/7, 3/11] Dec 2025 maksandchael Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
11 Hydra-like Cryptids BBf(23) too many to display - see page Aug 2026 Jason Yuen (@-d) and Claude Opus 4.6 Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting. Found "in the wild".
Hydra - compiled BBf(25) [363/14, 125/2, 22/21, 1/3, 7/11, 14/5] Dec 2025 maksandchael Compilation of Hydra.
BMO 1 - compiled BBf(36) [153/55, 2/11, 26/35, 3/7, 11/17, 7/13, 25/6, 55/2, 14/3] Nov 2025 maksandchael Compilation of BMO 1.
BBf(48) [77/2, 2/99, 17/33, 13/11, 285/119, 17/19, 1375/51, 1/17, 3/5, 243/7, 10/13] Nov 2025 maksandchael Similar to Space Needle, probviously non-halting.
BBλ(74) (λ1111(λ1)1(λλ1))(λλλ1(322222)) Apr 2025 50_ft_lock Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
BBCS(23) A++; B++; while B {while A {A--; C++; E++; while D {D--; E--;} while E {C++; D++; E--;}} C>>A; D>>B*3; B--;} Jun 2026 sheep Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
BB_PT(11) 11111_20_1 Jul 2026 sheep, Shawn Ligocki This machine halts iff 5 eventually converges to 1 under the 5n+3 map.
peBBle(12) 00_00_00_40_10_00_00_08_41_11_00_a1 Oct 2026 User:AlephSquirrel Similar to BMO 1.
peBBle(86) https://discord.com/channels/960643023006490684/1535695107448381591/1554751839721750569 Sep 2026 sheep Similar random walk mechanism to Hydra, Antihydra. Probviously non-halting.
MBB(8) 0+B_1-FC_2-GD_0-EF_1+C_2+A_2-EH_1-B* Oct 2026 Shawn Ligocki Similar to Space Needle, probviously non-halting. An equivalent machine was analyzed. Found "in the wild".
Needle 1 MBB(8) 0+B_1+C_2-AD_1-EH_2+F_1-DG_0-EB_2-C* Oct 2026 sheep Similar to Space Needle, probviously non-halting. Found "in the wild".
Hydra - compiled MBB(10) 0-BF_1+C_1+D_0-EH_1+A_2+G_2+I_2-I*_0+J_1-IA Jun 2025 Shawn Ligocki Compilation of Hydra.
5n+1 MBB(12,2) 0+B_0+C_0+D_0+E_1-IF_1+G_0-HE_0-FJ_0+A_1-KL_0+J_0-H* Oct 2026 sheep This machine halts iff 7 eventually converges to 1 under the 5n+1 map.
Erdős Ternary Conjecture MBB(19) 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* 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 MBB(129) https://www.sciencedirect.com/science/article/pii/S0304397519304633 Feb 2020 Yuri Matiyasevich The machine halts if and only if the Riemann hypothesis is false.
BB_boolf(24) *>*>*[>[*<<]*>*[<<]*>*>] Oct 2026 sheep Similar random walk mechanism to Bigfoot. Probviously non-halting. Translation of the smallest known BitChanger cryptid.
ZFC BBWT(11487) https://github.com/jconorgrogan/k230-wang-tiles-lean Oct 2026 jconorgrogan These Wang tiles tile the whole integer plane if and only if Zermelo–Fraenkel set theory with the Axiom of Choice is consistent. Not independently verified.

References

  1. ↑ https://github.com/lengyijun/goldbach_tm
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