Cryptids

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 domain	Machine	Announcement	Date	Discoverer	Note
Bigfoot	BB(3,3)	1RB2RA1LC_2LC1RB2RB_---2LA1LA (bbch)	BB(3, 3) is hard	Nov 2023	Shawn Ligocki
Hydra	BB(2,5)	1RB3RB---3LA1RA_2LA3RA4LB0LB0LA (bbch)	BB(2, 5) is hard	May 2024	Daniel Yuan
Antihydra	BB(6)	1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA (bbch)	Discord message	June 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.

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".

Name	BB domain	Machine	Announcement	Date	Discoverer	Note
ZF	BB(745)	O'Rear's machine: https://github.com/sorear/metamath-turing-machines/blob/master/zf2.nql Riebel's analysis: https://www.ingo-blechschmidt.eu/assets/bachelor-thesis-undecidability-bb748.pdf		2016, 2023	O’Rear and Riebel	The machine halts if and only if Zermelo–Fraenkel set theory is inconsistent.
RH	BB(744)	https://github.com/sorear/metamath-turing-machines/blob/master/riemann-matiyasevich-aaronson.nql		2016	Matiyasevich and O’Rear	The machine halts if and only if Riemann Hypothesis is false.
Goldbach	BB(25)	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		2016	anonymous	The machine halts if and only if Golbach's conjecture is false. Its behavior has been verified in Lean.
Erdős	BB(5,4) and BB(15)	https://docs.bbchallenge.org/other/powers_of_two_5_4.txt Machine code: 1RB3RA2RA1RB_0LC2RB1RA3RB_0LD1LC2LE3LC_3RE2RE---1RE_0RB1LE2LE3LE https://docs.bbchallenge.org/other/powers_of_two_15_2.txt Machine code: 1RB1RO_0RC0RC_0RD1RJ_0LE1RC_0LF1LK_0LG1LE_0LH1LF_1RI0LL_0RB1LK_1RC0RA_0LI1LN_1RM---_0RI0RO_0LK1LK_1LM1RA	arxiv preprint	Jul 2021	Tristan Stérin (@cosmo) and Damien Woods	The machine halts if and only if the following conjecture by Erdős is false: "For all n > 8, there is at least one 2 in the base-3 representation of 2^n"
Weak Collatz	BB(124) and BB(43,4)	https://docs.bbchallenge.org/other/weak_Collatz_conjecture_124_2.txt (unverified) https://docs.bbchallenge.org/other/weak_Collatz_conjecture_43_4.txt (unverified)		Jul 2021	Tristan Stérin	The machine halts if and only if the "weak Collatz conjecture" is false. The weak Collatz conjecture states that the iterated Collatz map (3x+1) has only one cycle on the positive integers. Not independently verified, and probably easy to further optimise.
Bigfoot - compiled	BB(7)	0RB1RB_1LC0RA_1RE1LF_1LF1RE_0RD1RD_1LG0LG_---1LB	Bigfoot Comment	June 2024	@Iijil1	Compilation of Bigfoot into 2 symbols, there was a previous compilation with 8 states
Hydra - compiled	BB(9)	0RB0LD_1LC0LI_1LD1LB_0LE0RG_1RF0RH_1RA---_0RD0LB_0RA---_0RF1RZ File:Hydra 9 states.txt	Discord message	June 2024	@Iijil1	Compilation of Hydra into 2 symbols, all confirmed by Shawn Ligocki. @Iijil1 provided 24 TMs which all emulate the same behavior. Previous compilation had 10 states, by Daniel Yuan, also confirmed by Shawn Ligocki.

Beeping Busy Beaver

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