BusyBeaverWiki - User contributions [en]

(Article title)

2026-05-10T21:50:09Z

C7X:

{{delete|Looks like things covered elsewhere on this wiki}}

La funzione bb(n) e incalcolabile, anche se, gli stati fanno crescere velocemente, ma i simboli con gli stati ancora di piu!, quindi dobbiamo fare computer con piu potenza di calcolo e altre cose!

SKI Calculus

2026-05-10T21:43:41Z

C7X: Replace with archived version

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

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

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

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

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

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

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

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

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

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

[[Category:Functions]]

Fast-Growing Hierarchy

2026-04-22T19:26:39Z

C7X: Spacing /* Examples */

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

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

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

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

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

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

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

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

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

*

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

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

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

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

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

Fast-Growing Hierarchy

2026-04-22T19:26:13Z

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

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

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

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

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

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

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

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

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

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

*

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

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

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

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

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

Fractran

2026-04-12T19:36:00Z

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

<div class="toccolours mw-collapsible mw-collapsed">'''Small Champions'''<div class="mw-collapsible-content">
{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
|-
| 2 || 1 || <code>[1/2]</code>
|<math>\begin{bmatrix}
-1
\end{bmatrix}</math>
|-
| 3 || 1 || <code>[3/2]</code>
|<math>\begin{bmatrix}
-1 & 1
\end{bmatrix}</math>
|-
| 4 || 1 || <code>[9/2]</code>
|<math>\begin{bmatrix}
-1 & 2
\end{bmatrix}</math>
|-
| 5 || 2 || <code>[3/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 1 \\
0 & -1
\end{bmatrix}</math>
|-
| 6 || 3 || <code>[9/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 2 \\
0 & -1
\end{bmatrix}</math>
|-
| 7 || 4 || <code>[27/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 3 \\
0 & -1
\end{bmatrix}</math>
|-
| 8 || 5 || <code>[81/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 4 \\
0 & -1
\end{bmatrix}</math>
|-
| 9 || 6 || <code>[243/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 5 \\
0 & -1
\end{bmatrix}</math>
|-
| 10 || 7 || <code>[729/2, 1/3]</code>
|<math>\begin{bmatrix}
-1 & 6 \\
0 & -1
\end{bmatrix}</math>
|-
| 11 || 10 || <code>[27/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 3 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 12 || 13 || <code>[81/2, 25/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 2 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 13 || 17 || <code>[81/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 4 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|-
| 14 || 21 || <code>[243/2, 125/3, 1/5]</code>
|<math>\begin{bmatrix}
-1 & 5 & 0 \\
0 & -1 & 3 \\
0 & 0 & -1
\end{bmatrix}</math>
|}
</div></div>

{| class="wikitable"
|+
!n
!BBf(n)
!Example Champion
!Vector Representation
!Champion Found
!Holdouts Proven
|-
| 15 || 28 || <code>[1/45, 4/5, 3/2, 25/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 2
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 16 || 53 || <code>[1/45, 4/5, 3/2, 125/3]</code>
|<math>\begin{bmatrix}
0 & -2 & -1 \\
2 & 0 & -1 \\
-1 & 1 & 0 \\
0 & -1 & 3
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434033599094587595 1 Nov 2025]
|-
| 17 || 107 || <code>[5/6, 49/2, 3/5, 40/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
3 & 0 & 1 & -1
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1434313398799175710 1 Nov 2025]
|Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1362008236118511758/1434771877376557086 3 Nov 2025]
|-
| 18 || 211 || <code>[5/6, 49/2, 3/5, 80/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
4 & 0 & 1 & -1
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1435313806493614131 4 Nov 2025]
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1362008236118511758/1436661215911870584 8 Nov 2025]
|-
| 19 || 370 || <code>[5/6, 49/2, 3/5, 160/7]</code>
|<math>\begin{bmatrix}
-1 & -1 & 1 & 0 \\
-1 & 0 & 0 & 2 \\
0 & 1 & -1 & 0 \\
5 & 0 & 1 & -1
\end{bmatrix}</math>
|@creeperman7002 [https://discord.com/channels/960643023006490684/1362008236118511758/1435763150489387090 5 Nov 2025]
|Decider: Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438558242388312165 13 Nov 2025]
3 Holdouts: Racheline & Shawn Ligocki
|-
|20
|746
|<code>[7/15, 22/3, 6/77, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 & 0 \\
1 & -1 & 0 & 0 & 1 \\
1 & 1 & 0 & -1 & -1 \\
-1 & 0 & 1 & 0 & 0 \\
0 & 2 & -1 & 0 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1438480761169776733 13 Nov 2025]
|Decider: Jason Yuen (@-d)
([https://github.com/int-y1/BBFractran/tree/main/holdout Enum+initial])
Daniel Yuan (@dyuan01) [https://discord.com/channels/960643023006490684/1438019511155691521/1438559507579011194 13] and [https://discord.com/channels/960643023006490684/1438019511155691521/1438996636389998773 14 Nov 2025]

Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1447069110541484146 7] and [https://discord.com/channels/960643023006490684/1438019511155691521/1453213088630444168 24 Dec 2025]
6 Holdouts: Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1452913055053778945 23 Dec 2025]
|-
|21
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1439759182587891894 16 Nov 2025]
|140 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 25 Jan 2026]
Claude Opus 4.6 proof of nonhalting of all 140: [https://discord.com/channels/960643023006490684/1438019511155691521/1485168251997786173 28 March 2026]
|-
|22
|<math>> 1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|<math>\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1448912286713384961 11 Dec 2025] and Jason Yuen (@-d)<sup>[https://discord.com/channels/960643023006490684/1438019511155691521/1448953682237460480 <nowiki>[1]</nowiki>]</sup>
|2003 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 25 Jan 2026]

Known [[Cryptid|Cryptids]]:

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

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

=== Behavior of Champions ===

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

This family obeys the following rules:

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

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

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

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

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

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

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

which follows the reasonably "lucky" trajectory:

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

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

The BBf(21) champion (running >31M steps):
<math display="block">\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>

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

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

==== BBf(22) ====
The BBf(22) champion (running <math>> 10^{62}</math> steps):
<math display="block">\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>

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

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

== Cryptids ==

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Start: A(7, 1)

A(1, b) --> halt

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

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

== References ==
<references />

[[Category:Functions]]

Cryptids

2026-04-09T05:38:58Z

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

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

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

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

== Cryptids at the Edge ==

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

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

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

== Larger Cryptids ==

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

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

== Beeping Busy Beaver ==

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

Cryptids

2026-04-09T05:37:22Z

C7X: Fix year /* Larger Cryptids */

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

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

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

== Cryptids at the Edge ==

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

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

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

== Larger Cryptids ==

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

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

== Beeping Busy Beaver ==

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

Cryptids

2026-04-09T05:37:07Z

C7X: Fix page numbers /* Larger Cryptids */

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

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

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

== Cryptids at the Edge ==

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

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

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

== Larger Cryptids ==

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

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

== Beeping Busy Beaver ==

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

Cryptids

2026-04-09T05:35:26Z

C7X: /* Larger Cryptids */

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

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

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

== Cryptids at the Edge ==

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

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

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

== Larger Cryptids ==

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

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

== Beeping Busy Beaver ==

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

Cryptids

2026-04-09T05:34:40Z

C7X: Add period /* Larger Cryptids */

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

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

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

== Cryptids at the Edge ==

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

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

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

== Larger Cryptids ==

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

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

== Beeping Busy Beaver ==

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

Cryptids

2026-04-09T05:34:17Z

C7X: /* Larger Cryptids */

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

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

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

== Cryptids at the Edge ==

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

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

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

== Larger Cryptids ==

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

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

== Beeping Busy Beaver ==

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

Graham's number

2026-03-14T18:36:55Z

C7X: Correction

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

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

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

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

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

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

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

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

Graham's number

2026-03-14T18:36:41Z

C7X: Update /* Bounds */

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

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

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

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

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

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

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

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

Graham's number

2026-03-14T18:36:08Z

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

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

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

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

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

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

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

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

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

Champions

2026-03-14T18:35:26Z

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

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

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

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

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

{| class="wikitable"
|+
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|[[BB(2)]]
|<math>6</math>
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
|[[Tibor Radó]]
|Direct Simulation
|-
|[[BB(3)]]
|<math>21</math>
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|Proven by [[Shen Lin]]
|Direct Simulation
|-
|[[BB(4)]]
|<math>107</math>
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|Allen Brady
|Direct Simulation
|-
|[[BB(5)]]
|<math>47\,176\,870</math>
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|Heiner Marxen & Jürgen Buntrock in 1989
|Direct Simulation
|-
|[[BB(6)]]
|<math>> 2\uparrow\uparrow\uparrow 5</math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|mxdys in 2025
|See mxdys's analysis on the TM page
|-
|[[BB(7)]]
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|[https://discord.com/channels/960643023006490684/1369339127652159509/1370678203395604562 Pavel Kropitz in 2025]
|Analyzed by Shawn Ligocki (see TM page)
|-
|[[BB(8)]]
|
|
|
|
|-
|BB(9)
|<math>> f_\omega(f_9(2))</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|Jacobzheng in 2024
|
|-
|BB(10)
|<math>> f_\omega^2(25)</math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|Racheline in 2024
|
|-
|BB(11)
|<math>> f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9))</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE|halt}}
|Racheline in 2024
|
|-
|BB(12)
|<math>> f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2))</math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|Racheline in 2024
|
|-
|BB(13)
|<math>> f_{\omega + 1}(2047) > g_{64}</math>
|{{TM|1RB1RA_1LC1RD_1LA1LC_1LG0RE_1LC1RB_0RL1LG_0LM0RH_1RI1RH_1LK0RI_---0LK_1LF1LK_1LJ1RL_1RZ1RH|halt}}
|[https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 50_ft_lock in 2026]
|
|-
|BB(14)
|<math>> f_{\omega + 1}(65\,536)</math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Racheline in 2024]
|
|-
|BB(15)
|<math>> f_{\omega + 1}(f_\omega(10^{57}))</math>
|{{TM|0RH1LD_1RI0RC_1RB1LD_0LD1LE_1LF1RA_1RG0LE_1RB1RG_1RD1RA_0LN0RJ_1RZ0LK_0LK1LL_1RG1LM_0LL0LL_1LO1LN_0LG1LN|halt}}
|Jacobzheng in 2025
|
|-
|BB(16)
|<math>> f_{\omega + 1}^2(10^{10^{57}})</math>
|[[User:Jacobzheng/BB(16)]]
|Jacobzheng in 2025
|
|-
|BB(17)
|
|
|
|
|-
|BB(18)
|<math>> f_{\omega + 2}(f_{\omega + 1}^3(f_{\omega}^2(60)))</math>
|[[User:Jacobzheng/BB(18)]]
|Jacobzheng in 2025
|
|-
|BB(19)
|
|
|
|
|-
|BB(20)
|<math>> f_{\omega + 2}^2(21)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274414683331366924 Racheline in 2024]
|
|-
|BB(21)
|<math>> f_{\omega^2}^2(4 \uparrow\uparrow 341)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|
|-
|BB(40)
|<math>> f_{\omega^\omega}(75\,500)</math>
|[[User:Jacobzheng/BB(40)]]
|Jacobzheng in 2024
|
|-
|BB(41)
|<math>> f_{\omega^\omega}^4(32)</math>
|[[User:Jacobzheng/BB(41)]]
|Jacobzheng in 2024
|
|-
|BB(51)
|<math>> f_{\varepsilon_0 + 1}(8)</math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Racheline in 2024]
|
|-
|BB(150)
|<math>f_{lim(BMS)}(10\uparrow\uparrow 15)</math>
|[https://morphett.info/turing/turing.html?c95a199c8e8a3dd56452f8b7e28fabbf too large to show]
|Patcail in 2025<ref>https://discord.com/channels/960643023006490684/1026577255754903572/1328863966688182345</ref>
|
|}

== 3-Symbol TMs ==

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

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

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

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

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

== References ==

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

Logical independence

2026-03-14T18:34:13Z

C7X: More common phrasing /* Large cardinals */

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== References ==
<references />

[[Category:Zoology]]

BB(7)

2026-03-14T08:50:44Z

C7X: Done /* History */

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

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

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

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

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

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

[[Probviously]] non-halting Cryptids:

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

== Top Halters ==
The scores are given using [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]] with an extension to decimal tetration<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>. The top 20 scoring known machines are:
{| class="wikitable sortable"
!TM
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|data-sort-value="10 ↑↑ 9999"|<math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|Pavel Kropitz
|-
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE|halt}}
|10 ↑↑ 519.20
|Andrew Ducharme
|-
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE|halt}}
|10 ↑↑ 519.20
|Andrew Ducharme
|-
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD0LG_1LB0LE|halt}}
|10 ↑↑ 519.20
|@gerbil5709, Terry Ligocki
|-
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG|halt}}
|10 ↑↑ 403.84
|Andrew Ducharme
|-
|{{TM|1RB1RF_0RC1RG_1LD1LE_0LE1LD_0RF0LC_1RA0LC_0RF1RZ|halt}}
|10 ↑↑ 286.17
|Terry Ligocki
|-
|{{TM|1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA0LG_1RZ0RD|halt}}
|10 ↑↑ 246.32
|@Iijil
|-
|{{TM|1RB0LE_1RC0RA_1RD0RC_1LE1LD_1LA0LF_0LA1LG_1RZ1LA|halt}}
|10 ↑↑ 246.32
|@star, Terry Ligocki
|-
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG|halt}}
|10 ↑↑ 243.88
|@Iijil, Andrew Ducharme
|-
|{{TM|1RB0RB_1LC1RG_1RD1RC_1RE0RA_1LF0LB_1RF0LE_0RD1RZ|halt}}
|10 ↑↑ 228.78
|Terry Ligocki
|-
|{{TM|1RB0LD_0LC1RZ_1RA0RD_1RE1LD_1LF0RC_0LG1LE_1RG0LD|halt}}
|10 ↑↑ 192.67
|Terry Ligocki
|-
|{{TM|1RB1LA_1LC0RE_0LD1LB_1RD0LA_1RF0RA_0RG0LA_1RB1RZ|halt}}
|10 ↑↑ 192.67
|Terry Ligocki
|-
|{{TM|1RB1LA_1LC0RE_0LD1LB_1RD0LA_1RF0RA_1RG0LA_0LE1RZ|halt}}
|10 ↑↑ 192.67
|Terry Ligocki
|-
|{{TM|1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RA0LG_1RB1LG|halt}}
|10 ↑↑ 192.67
|Terry Ligocki
|-
|{{TM|1RB1RZ_1LC0RE_0LD1LB_1RE0LA_1RF0RG_0RG0LG_1RB1LG|halt}}
|10 ↑↑ 192.67
|Terry Ligocki
|-
|{{TM|1RB0LD_0LC1RZ_1RA0RD_1RE1LD_1LF0RC_0LG1LE_1RC0LD|halt}}
|10 ↑↑ 192.67
|Andrew Ducharme
|-
|{{TM|1RB1LA_1LC0RE_0LD1LB_1RE1LG_1RF0LG_0RA0LA_0RF1RZ|halt}}
|10 ↑↑ 192.67
|Andrew Ducharme
|-
|{{TM|1RB1LA_1LC0RF_0LD0RD_1RF1LE_1LB1RZ_1RG0RA_0RA0LA|halt}}
|10 ↑↑ 192.67
|@C7X
|-
|{{TM|1RB1LA_1LC0RE_0LD1LB_1RE0LA_1RF0RA_0RG0LA_1RB1RZ|halt}}
|10 ↑↑ 192.67
|@Iijil, Terry Ligocki
|-
|{{TM|1RB1LA_1LC0RE_0LD1LB_1RE0LA_1RF0RA_1RG0LA_0LE1RZ|halt}}
|10 ↑↑ 192.67
|@Iijil, Terry Ligocki
|}

The top 20 known halters with unique scores are:

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

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

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

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

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

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

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

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

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Runtime (hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

{| class="wikitable sortable" style="text-align: right"
! style="width: 10%" rowspan="2" |Done by
! colspan="2" |Holdout TMs
! rowspan="2" |<math>*^1</math>
! rowspan="2" |<math>*^2</math>
! colspan="2" |TMs/sec/core
! style="width: 50%" rowspan="2" |Description
! rowspan="2" |Source
|-

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

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

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

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

(done to reduce column size:
<math>*^1</math>= % Reduced,
<math>*^2</math>= Compute Time (core-hours),
<math>*^3</math>= Decided,
<math>*^4</math>= Processed)

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

Graham's number

2026-03-14T08:15:06Z

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

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

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

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

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

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

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

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

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

Graham's number

2026-03-14T07:52:55Z

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

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

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

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

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

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

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

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

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

Graham's number

2026-03-14T07:43:10Z

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

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

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

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

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

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

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

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

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

Logical independence

2026-03-01T05:22:05Z

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== References ==
<references />

[[Category:Zoology]]

Talk:BB(11)

2026-02-28T02:43:17Z

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

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

Logical independence

2026-02-28T02:40:45Z

C7X: Add to citations /* Large cardinals */

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== References ==
<references />

[[Category:Zoology]]

Logical independence

2026-02-28T02:36:29Z

C7X: MathJax

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== References ==
<references />

[[Category:Zoology]]

Logical independence

2026-02-28T02:34:27Z

C7X: Infinitely many axioms

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

== References ==
<references />

[[Category:Zoology]]

Talk:Logical independence

2025-11-06T07:59:45Z

C7X: /* Extensionality */

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

== ZF vs. ZFC ==

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

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

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

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

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

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

Talk:Logical independence

2025-11-06T07:59:16Z

C7X: /* Extensionality */

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

== ZF vs. ZFC ==

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

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

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

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

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

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

Talk:Logical independence

2025-11-06T07:58:53Z

C7X: /* ZF vs. ZFC */

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

== ZF vs. ZFC ==

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

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

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

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

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

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

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

2025-10-12T18:26:24Z

C7X:

{{Stub}}

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

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

Cryptids

2025-09-03T15:27:36Z

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

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

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

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

== Cryptids at the Edge ==

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

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

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

== Larger Cryptids ==

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

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

== Beeping Busy Beaver ==

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

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

2025-08-31T03:50:53Z

C7X: /* More simulation */

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

== More simulation ==

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

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

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

2025-08-30T11:16:49Z

C7X: /* More simulation */

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

2025-08-30T11:16:02Z

C7X: /* More simulation */ new section

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

== More simulation ==

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

Maximum Consecutive Ones Function

2025-08-15T01:06:04Z

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

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

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

== Champions ==

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

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

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

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

BB(7)

2025-08-02T08:00:28Z

C7X: Add 48xxxxx machine /* Top Halters */

Talk:Logical independence

2025-07-22T00:07:44Z

C7X: Simplify /* ZF vs. ZFC */

Talk:Logical independence

2025-07-22T00:07:06Z

C7X: /* ZF vs. ZFC */

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

== ZF vs. ZFC ==

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

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

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

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

Talk:Logical independence

2025-07-21T20:04:53Z

C7X: /* ZF vs. ZFC */

BB(7)

2025-07-17T01:32:33Z

C7X: Number of phase 1 holdouts instead /* Current Progress */

BB(7)

2025-07-17T01:26:06Z

C7X: Done. (3,084,940 machines undecided after linear rule) /* Current Progress */

BB(7)

2025-07-11T17:19:11Z

C7X: More linear /* Current Progress */

BB(7)

2025-07-02T04:39:32Z

C7X: More /* Current Progress */

BB(7)

2025-07-01T18:29:06Z

C7X: Add unclaimed blocks /* Current Progress */

BB(7)

2025-07-01T18:22:40Z

C7X: Adding usernames which got lost in last edit /* Current Progress */

BB(6)

2025-06-28T07:46:40Z

C7X: Update /* Top Halters */

The 6-state, 2-symbol Busy Beaver problem, '''BB(6)''', refers to the unsolved 6<sup>th</sup> value of the [[Busy Beaver function]]. With the discovery of the [[Cryptid]] machine [[Antihydra]] in June 2024, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(6) and thus [https://www.sligocki.com/2024/07/06/bb-6-2-is-hard.html BB(6) is Hard].

The current BB(6) champion {{TM|1RB1RA_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

<math display="block">S(6) > \Sigma(6) > 2 \uparrow\uparrow\uparrow 5</math>

== Techniques ==
Simulating tetrational machines, such as the former champion {{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}, requires [[Accelerated simulator|accelerated simulation]] that can handle Collatz Level 2 [[Inductive rule|inductive rules]]. In other words, it requires a simulator that can prove the rules:

<math display="block">\begin{array}{lcl}
C(4k) & \to & {\operatorname{Halt}}\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+1) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+2) & \to & C\Big(\frac{3^{k+3} - 11}{2}\Big) \\
C(4k+3) & \to & C\Big(\frac{3^{k+3} + 1}{2}\Big) \\
\end{array}</math>

and also compute the remainder mod 3 of numbers produced by applying these rules 15 times (which requires some fancy math related to [[wikipedia:Euler's_totient_function|Euler's totient function]]).

== Cryptids ==
Several [[Turing machines]] have been found that are [[Cryptids]], considered so because each of them have a [[Collatz-like]] halting problem, a type of problem that is generally difficult to solve. However, probabilistic arguments have allowed all but one of them to be categorized as [[probviously]] halting or probviously non-halting.

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

There are a few machines considered notable for their chaotic behaviour, but which have not been classified as Cryptids due to seemingly lacking a connection to any known open mathematical problems, such as Collatz-like problems.

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD}}

== Top Halters ==
Below is a table of the machines with the 10 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of Knuth's up-arrow notation.<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE}}
|10 ↑↑ 15.60465
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA}}
|10 ↑↑ 7.23619
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB}}
|10 ↑↑ 6.96745
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE}}
|10 ↑↑ 5.77573
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD}}
|10 ↑↑ 5.63534
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Holdouts ==
@mxdys's informal [[Holdouts lists|holdouts list]] has 3335 machines up to equivalence as of June 2025.

== References ==
<references />
[[Category:BB Domain]]

BB(7)

2025-06-27T16:26:33Z

C7X: /* Top Halters */

BB(7)

2025-06-27T16:21:50Z

C7X: /* Current Progress */

BB(7)

2025-06-27T16:18:41Z

C7X: Some linear rule /* Current Progress */

Lucy's Moonlight

2025-06-26T12:02:23Z

C7X: Update champion

{{machine|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}{{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}

'''Lucy's Moonlight''' is a [[probviously]] halting tetrational [[BB(6)]] [[Cryptid]] found by Racheline on 1 Mar 2025 ([https://discord.com/channels/960643023006490684/1239205785913790465/1345551751016878272 Discord link]). It has a 5% chance of beating the former BB(6) champion {{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE}} (if we treat Collatz-like behavior as random).

== Analysis by Racheline ==
https://discord.com/channels/960643023006490684/1345810396136865822/1345820781363597312<pre>
A(x,y) := 0^inf (1011)^x 10 <A (01)^y 0^inf
B(x) := 0^inf 1^x B> 0^inf

A(x+1,3y) -> A(x,8y+3)
A(x+2,3y+1) -> A(x,8y+11)
A(x+2,3y+2) -> A(x,8y+12)
A(0,y) -> B(2y+1)
A(1,3y+1) -> A(4y+4,4)
A(1,3y+2) -> halt
B(3y) -> B(8y-2)
B(3y+1) -> A(2y,4)
B(3y+2) -> B(8y+6)

a is the sequence such that A(x,a_n) goes to A(x',a_(n+1)) in one step assuming x>=2
b is the sequence such that A(x,a_0) goes to A(x-b_n,a_n) in n rules (without using the A(0,y) or A(1,y) rules) assuming x>=b_n
c is the sequence such that A(c_n,a_0) goes to A(c_(n+1),a_0) after only one application of the A(0,y) or A(1,y) rules

f(3n) = 8n+3
f(3n+1) = 8n+11
f(3n+2) = 8n+12
a_0 = 4
a_(n+1) = f(a_n)
b_0 = 0
b_(n+1) = b_n+(1 if 3|a_n else 2)

g(n) = 8ceil(n/3)-2 (i.e. g(3n) = 8n-2 and g(3n+2) = 8n+6)
h(n) = largest i such that b_i <= n
c_0 = 14
if b_h(c_n) = c_n, then c_(n+1) = 2m where m is minimal such that 3m+1 = g^k(2a_h(c_n)+1) for some k
otherwise if a_h(c_n) = 3m+1, then c_(n+1) = 4m+4
otherwise halt after roughly (8/3)^(6c_n/5) steps
</pre>

== Analysis by Shawn Ligocki ==
https://discord.com/channels/960643023006490684/1345810396136865822/1346329322851401868<pre>
1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA

C(a, b, c) = 0^inf 1011^a 1^b 10^c C> 0^inf
C(a, b) = C(a, b, 1) = 0^inf 1011^a 1^b 10 C> 0^inf

C(a+1, 3k) --> C(a, 8k+6)
C(a+2, 3k+1) --> C(a, 8k+16)
C(a+2, 3k+2) --> C(a, 8k+22)

C(0, 3k) --> C(2k, 8)
C(0, 3k+1) --> C(0, 8k+5)
C(1, 3k+1) --> Halt(6k+14)
C(0, 3k+2) --> C(0, 8k+5)
C(1, 3k+2) --> C(2k+4, 8)

Start --(2)--> C(0, 0)
</pre>Racheline's <math>c_n</math> values correspond to the values at which we get configs <math>C(c_n, 8)</math> in my analysis. The first three are:

* <math>c_0 = 14</math>
* <math>c_1 = 11\,292</math>
* <math>c_2 = </math> 8 282 581 182 265 963 777 660 116 067 041 084 396 825 871 729 769 475 063 015 437 507 606 888 488 657 640 984 741 410 755 868 651 202 413 557 949 100 792 150 345 468 805 096 096 950 985 621 014 344 543 514 277 244 259 988 659 130 143 328 155 990 590 108 709 713 794 583 253 323 686 355 356 512 219 061 229 636 197 885 927 258 835 226 571 319 297 308 352 230 934 484 006 197 639 625 592 087 971 234 386 001 742 614 119 317 946 288 524 516 349 575 343 597 522 485 283 906 000 542 032 088 582 043 646 950 532 639 630 385 985 319 217 159 379 913 000 142 879 141 905 099 969 565 530 694 702 807 960 713 276 894 845 659 927 803 312 770 155 623 893 892 127 451 599 296 902 730 174 376 009 710 735 758 161 389 656 270 797 836 582 256 488 031 353 066 716 635 172 987 950 448 854 471 226 597 449 927 236 184 172 841 640 111 209 332 317 049 722 869 659 569 874 196 714 141 959 835 401 796 418 444 068 891 026 981 841 656 732 128 708 017 637 486 218 786 090 173 524 036 425 924 718 502 564 851 924 717 340 390 259 248 282 032 112 075 387 681 859 362 344 399 913 313 735 645 684 525 131 229 468 282 784 360 728 881 748 147 372 112 747 036 418 378 308 364 410 128 040 328 676 209 420 026 633 482 346 143 509 117 105 276 670 245 493 297 604 407 287 182 199 289 609 254 900 080 171 095 368 953 306 931 467 191 729 590 199 363 109 109 618 828 683 456 945 716 771 345 293 252 204 756 902 270 830 478 266 505 243 340 324 828 877 091 406 917 371 244 363 787 314 164 920 400 219 556 757 173 398 748 668 149 395 792 060 530 400 633 872 912 079 249 392 256 126 285 748 793 796 259 657 854 699 829 517 626 609 309 417 076 213 461 174 150 922 612 299 942 658 509 909 739 815 101 078 137 303 456 289 178 147 820 849 027 886 955 738 533 503 625 157 087 287 391 831 669 455 397 075 444 062 908 165 633 623 616 230 849 011 917 173 994 535 718 598 409 770 737 638 239 724 998 256 861 644 166 630 982 723 063 781 225 891 358 955 048 633 229 232 741 851 699 498 876 266 677 026 907 578 098 686 784 323 407 335 765 343 701 077 746 445 802 114 304 942 791 377 887 303 588 107 097 550 867 703 017 440 867 027 391 593 474 985 628 594 939 344 739 091 341 577 631 399 711 011 114 031 231 392 355 268 858 286 239 590 222 739 798 802 836 719 470 359 138 334 346 097 704 505 528 574 751 020 940 898 407 003 617 333 219 550 156 008 932 231 022 648 658 161 473 903 774 681 072 952 056 320 551 244 912 271 864 381 014 835 634 282 966 523 463 985 953 949 176 576 786 408 020 337 836 220 233 538 960 379 978 664 849 564 796 907 967 238 406 785 655 383 464 715 057 232 716 549 617 630 853 473 306 701 852 904 885 606 863 527 445 121 198 511 795 947 547 473 084 736 520 465 867 328 457 114 967 373 145 664 842 684 299 186 357 472 668 934 663 435 131 859 827 025 004 311 417 517 867 339 721 854 899 421 107 496 540 906 158 691 502 108 225 108 606 878 323 722 157 711 420 937 080 640 504 487 833 073 951 122 223 191 857 376 281 164 771 129 574 147 051 021 933 227 085 689 827 336 672 385 780 462 357 440 219 398 403 103 894 053 177 742 661 833 002 598 347 813 276 596 599 779 500 262 101 470 481 321 560 131 349 255 581 918 937 061 811 724 247 415 920 101 784 002 187 650 019 023 113 164 778 491 907 065 101 691 880 663 970 914 566 257 787 660 257 139 627 941 284 081 248 034 057 979 564 419 991 961 345 637 080 391 063 343 278 659 875 006 722 945 435 258 511 750 736 395 209 297 602 991 695 886 580 794 249 759 902 536 380 055 710 080 977 728 337 952 498 718 259 606 078 790 355 738 625 858 597 516 649 364 990 083 397 742 948 406 390 695 711 840 139 170 928 984 527 396 432 236 103 181 667 418 006 635 667 189 873 871 634 905 950 683 958 299 923 219 653 264 060 399 588 571 782 767 511 747 924 043 969 623 045 308 763 567 170 166 295 093 013 227 497 346 173 854 101 964 306 147 690 034 349 284 712 163 842 269 859 435 320 408 715 901 523 168 064 489 459 605 434 861 066 611 056 209 626 985 578 507 010 825 633 753 454 <math>\approx 10^{2901.92}</math>

== See Also ==

* [https://www.sligocki.com/2025/04/21/lucys-moonlight.html Lucy's Moonlight: The 5% Champion]. Shawn Ligocki. Apr 2025.

[[Category:Cryptids]]

Champions

2025-06-26T11:58:42Z

C7X: /* 2-Symbol TMs */

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

== 2-Symbol TMs ==
Rows are blank if no champion has been found which surpasses a smaller size problem. Take also note that the <math> f_{x}(n) </math> used in the lowerbounds represent the [[Fast-Growing Hierarchy]]. Note that most champions above 6 states are self-reported and have not been independently verified.

{| class="wikitable"
|+
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|[[BB(2)]]
|<math> 6 </math>
|{{TM|1RB1LB_1LA1RZ|halt}} {{TM|1RB0LB_1LA1RZ|halt}} {{TM|1RB1RZ_1LB1LA|halt}} {{TM|1RB1RZ_0LB1LA|halt}} {{TM|0RB1RZ_1LA1RB|halt}}
|Tibor Radó
|Direct Simulation
|-
|[[BB(3)]]
|<math> 21 </math>
|{{TM|1RB1RZ_1LB0RC_1LC1LA|halt}}
|Proven by Shen Lin
|Direct Simulation
|-
|[[BB(4)]]
|<math> 107 </math>
|{{TM|1RB1LB_1LA0LC_1RZ1LD_1RD0RA|halt}}
|Allen Brady
|Direct Simulation
|-
|[[BB(5)]]
|<math> 47\,176\,870 </math>
|{{TM|1RB1LC_1RC1RB_1RD0LE_1LA1LD_1RZ0LA|halt}}
|Heiner Marxen & Jürgen Buntrock in 1989
|Direct Simulation
|-
|[[BB(6)]]
|<math> > 2\uparrow\uparrow\uparrow 5 </math>
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|mxdys in 2025
|See mxdys's analysis on the TM page
|-
|[[BB(7)]]
|<math>> 2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|{{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}
|[https://discord.com/channels/960643023006490684/1369339127652159509/1370678203395604562 Pavel Kropitz in 2025]
|Analyzed by Shawn Ligocki (see TM page)
|-
|BB(8)
|
|
|
|
|-
|BB(9)
|<math> > f_\omega(f_9(2)) </math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_1LB0LH|halt}}
|Jacobzheng in 2024
|
|-
|BB(10)
|<math> > f_\omega^2(25) </math>
|{{TM|1RB1RA_0LC0LF_0RD1LC_1RA1RG_1RZ0RA_1LB1LF_1LH1RE_0LI1LH_0LF0LJ_1LH0LJ|halt}}
|Racheline in 2024
|
|-
|BB(11)
|<math> > f_\omega^2(2 \uparrow\uparrow 12) > f_\omega^2(f_3(9)) </math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RZ0LI_0LD1LE|halt}}
|Racheline in 2024
|
|-
|BB(12)
|<math> > f_\omega^4(2 \uparrow\uparrow\uparrow 4-3) > f_\omega^4(f_4(2)) </math>
|{{TM|0LJ0RF_1LH1RC_0LD0LG_0RE1LD_1RF1RA_1RB1RF_1LC1LG_1LL1LI_1LK0LH_1RH1LJ_1RZ1LA_1RF1LL|halt}}
|Racheline in 2024
|
|-
|BB(13)
|
|
|
|
|-
|BB(14)
|<math> > f_{\omega + 1}(65\,536) > g_{64} </math>
|{{TM|1LH1LA_1LI1RG_0RD1LC_0RF1RE_1LJ0RF_1RB1RF_0LC1LH_0LC0LA_1LK1LJ_1RL0LI_0LL1LE_1LM1RZ_0LN1LF_0LJ---|halt}}
|[https://discord.com/channels/960643023006490684/960643023530762341/1274366178529120287 Racheline in 2024]
|
|-
|BB(15)
|
|
|
|
|-
|BB(16)
|<math> > f_{\omega + 1}(2 \uparrow\uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow\uparrow 9) </math>
|
|Daniel Nagaj in 2021
|[https://www.sligocki.com/2022/07/11/bb-16-graham.html Analysis by Shawn Ligocki]
|-
|BB(17)
|<math> > f_{\omega + 1}(f_\omega(60)) </math>
|[[User:Jacobzheng/BB(17)]]
|Jacobzheng in 2024
|
|-
|BB(18)
|<math> > f_{\omega + 1}(f_\omega^2(60)) </math>
|[[User:Jacobzheng/BB(18)]]
|Jacobzheng in 2024
|
|-
|BB(19)
|<math> > f_{\omega + 1}^3(f_\omega(60)) </math>
|[[User:Jacobzheng/BB(19)]]
|Jacobzheng in 2024
|
|-
|BB(20)
|<math> > f_{\omega + 2}^2(21) </math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274414683331366924 Racheline in 2024]
|
|-
|BB(21)
|<math> > f_{\omega^2}^2(4 \uparrow\uparrow 341) </math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1274471360206344213 Racheline in 2024]
|
|-
|BB(40)
|<math> > f_{\omega^\omega}(75\,500) </math>
|[[User:Jacobzheng/BB(40)]]
|Jacobzheng in 2024
|
|-
|BB(41)
|<math> > f_{\omega^\omega}^4(32) </math>
|[[User:Jacobzheng/BB(41)]]
|Jacobzheng in 2024
|
|-
|BB(51)
|<math> > f_{\varepsilon_0 + 1}(8) </math>
|
|[https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495 Racheline in 2024]
|
|}

== 3-Symbol TMs ==

{| class="wikitable"
|+
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|[[BB(2,3)]]
|<math> 38 </math>
|{{TM|1RB2LB1RZ_2LA2RB1LB|halt}}
|
|
|-
|[[BB(3,3)]]
|<math> > 10^{17} </math>
|{{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}}
|
|
|-
|[[BB(4,3)]]
|<math> > 2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|
|
|}

== 4-Symbol TMs ==
{| class="wikitable"
|+
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|BB(2,4)
|<math> 3\,932\,964 </math>
|{{TM|1RB2LA1RA1RA_1LB1LA3RB1RZ|halt}}
|Shawn & Terry Ligocki in 2005
|Pascal Michel, Heiner Marxen, Allen Brady
|-
|BB(3,4)
|<math> > 2 \uparrow^{15} 5 </math>
|{{TM|1RB3LB1RZ2RA_2LC3RB1LC2RA_3RB1LB3LC2RC|halt}}
|
|
|}

== 5-Symbol TMs ==
{| class="wikitable"
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|BB(2,5)
|<math> > 10^{10^{10^{3\,314\,360}}} </math>
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|
|
|-
|BB(3,5)
|<math> > f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15) </math>
|{{TM|1RB3LB4LC2RA4LB_2LC3RB1LC2RA1RZ_3RB1LB3LC2RC4LC|halt}}
|
|
|}

== 6-Symbol TMs ==
{| class="wikitable"
!
!Runtime
!Champions
!Discovered By
!Verification
|-
|BB(2,6)
|<math> > 10 \uparrow\uparrow 10 \uparrow\uparrow 10^{10^{115}} </math>
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|
|
|}