BusyBeaverWiki - User contributions [en]

SKI Calculus

2026-05-09T23:07:17Z

ADucharme: introduction reorganization

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://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]]

Busy Beaver for SKI calculus

2026-05-09T23:04:50Z

ADucharme: ADucharme moved page Busy Beaver for SKI calculus to SKI Calculus: standardization

#REDIRECT [[SKI Calculus]]

SKI Calculus

2026-05-09T23:04:50Z

ADucharme: ADucharme moved page Busy Beaver for SKI calculus to SKI Calculus: standardization

Busy Beaver for SKI calculus (we will call it BB_SKI for now) is a variation of the [[Busy Beaver for lambda calculus|Busy Beaver problem for lambda calculus]].

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.

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://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]]

Lambda Calculus

2026-05-09T23:02:00Z

ADucharme: /* See Also */ add De Bruijn index, edit SKI calculus links

'''Lambda calculus''' is a model of computation developed by Alonzo Church (of Church-Turing thesis fame) in 1936. If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with [[wikipedia:Lambda_calculus|this article]].

'''BBλ''' is the [[Busy Beaver]] problem for lambda calculus, where BBλ(n) is the maximum normal form size of any closed lambda term of size n (or 0 if no closed term of size n exists). Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus], a binary prefix-free encoding for all closed lambda calculus terms. Pioneered by John Tromp, BBλ is uncomputable, and therefore grows faster than any computable function.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if any choice of beta-reductions will lead to this normal form and '''weakly normalizing''' if there exist divergent reduction paths which never reach the normal form.

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) = 0 for those values.
<math>C(n)</math> denotes Church numeral ''n'' = <math>\lambda f\lambda x. f^n(x)</math>.
In the last column, JT and BF abbreviate John Tromp and Bertram Felgenhauer. The [https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 smallest Cryptid known] currently is in 74 bits.

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
|4 || = 4 || <math>\lambda 1</math>
||| JT & BF
|-
|6 || = 6 || <math>\lambda\lambda 1</math>
||| JT & BF
|-
|7 || = 7 || <math>\lambda\lambda 2</math>
||| JT & BF
|-
|8 || = 8 || <math>\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|9 || = 9 || <math>\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|10 || = 10 || <math>\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|11 || = 11 || <math>\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|12 || = 12 || <math>\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|13 || = 13 || <math>\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|14 || = 14 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|15 || = 15 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|16 || = 16 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|17 || = 17 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|18 || = 18 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|19 || = 19 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|20 || = 20 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|21 || = 22 || <math>\lambda(\lambda 1 1) (1 (\lambda 2))</math>
| <math>\lambda(1(\lambda 2))(1(\lambda 2))</math>|| JT & BF
|-
|22 || = 24 || <math>\lambda(\lambda 1 1 1) (1 1)</math>
| <math>\lambda(1 1) (1 1) (1 1)</math>|| JT & BF
|-
|23 || = 26 || <math>\lambda(\lambda 1 1) (1 (\lambda\lambda 2))</math>
| <math>\lambda(1 (\lambda\lambda 2)) (1 (\lambda\lambda 2))</math>|| JT & BF
|-
|24 || = 30 || <math>\lambda(\lambda 1 1 1) (1 (\lambda 1))</math>
| <math>\lambda(1 (\lambda 1)) (1 (\lambda 1)) (1 (\lambda 1))</math>|| JT & BF
|-
|25 || = 42 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|26 || = 52 || <math>(\lambda 1 1) (\lambda\lambda 2 (1 2))</math>
| <math>\lambda\lambda 2 (\lambda\lambda 2 (1 2)) (1 (2 (\lambda\lambda 2 (1 2))))</math>|| JT & BF
|-
|27 || = 44 || <math>\lambda\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|28 || = 58 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 (\lambda 2))))</math>
| <math>\lambda 1 (\lambda\lambda 1 (3 (\lambda 2))) (1 (\lambda 2 (\lambda\lambda 1 (4 (\lambda 2)))))</math>|| JT & BF
|-
| 29 || = 223|| <math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 1)))</math>
| <math>\lambda B (B (1 B)) \text{ where}</math>
<math>B = (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (2 1))))</math>
||JT & BF
|-
|30
|= 160
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (1 2))</math>
|
<math>\lambda\lambda 2 B A (1 (2 B A)) \text{ where}</math>
<math>B = (\lambda\lambda 2 A (1 (2 A)))</math>,
<math>A = (\lambda\lambda 2 (1 2))</math>
|JT & BF
|-
|31
|= 267
|<math>(\lambda 1 1) (\lambda\lambda 2 (2 (1 2)))</math>
|
<math>\lambda\lambda 2 A (2 A (C (2 A))) \text{ where}</math>
<math>C = (2 A (2 A (1 B (2 A))))</math>,
<math>B = (\lambda 3 A (3 A (1 (3 A))))</math>,
<math>A = (\lambda\lambda 2 (2 (1 2)))</math>
|JT & BF
|-
|32
|= 298
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 (\lambda 2))))</math>
|
|JT & BF
|-
|33
|= 1812
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (1 (2 1))))</math>
|
<math>\lambda C (C (C (1 C))) \text{ where}</math>
<math>C = (B (B (B (1 B)))</math>,
<math>B = (A (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (1 (2 1)))))</math>
|JT & BF
|-
|34 || <math>= 327\,686</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^2}})</math>|| JT & BF
|-
|35 || <math>= 5 \cdot 3^{3^3} + 6</math>
<math> > 3.8 \times 10^{13}</math>
| <math>(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^3})</math>|| JT & BF
|-
|36 || <math>= 5 \cdot 2^{2^{2^3}} + 6</math>
<math> > 5.7 \times 10^{77}</math>
| <math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 2 (2 1))))</math>
| <math>C(2^{2^{2^3}})</math>|| JT & BF
|-
|37 || <math> = 2 + BB\lambda(35)</math>
| <math>\lambda(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>\lambda x. C(3^{3^3})</math>||mxdys & JT & dyuan & sligocki
|-
|38 || <math>= 5\cdot{2^{2^{2^{2^2}}}} + 6</math>
| <math>(\lambda 1 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^{2^2}}})</math>|| JT & BF & CppDS & mxdys & sligocki & dyuan & charles
|-
|39 || <math>\ge 10^{10^{12}}</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{3^3}})</math>|| JT & BF
|-
|40 || <math> > 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{15} 33</math>
|| mxdys & racheline
|-
|41 || <math>\ge 10^{10^{40}}</math>
| <math>(\lambda 1 (\lambda 1 1) 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge 2 + BB\lambda(40)</math>
|<math>\lambda(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math>,
<math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge 2 + BB\lambda(43)</math>
| <math>\lambda(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| ||
|-
|46 || <math> \ge 2 + BB\lambda(44)</math>
| <math>\lambda(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|47 || <math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
| <math>(\lambda 1 1 1)(\lambda\lambda 1 (1 2) (\lambda\lambda 2 (2 1)))</math>
| ||50_ft_lock
|-
|48 || <math> > 10 \uparrow\uparrow\uparrow\uparrow 4</math>
| <math>(\lambda 1 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math> > Graham's number
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 1 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right) )</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<math>(\lambda 1 1 1) (\lambda 1 (1 (\lambda\lambda\lambda 1 3 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right) )</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|<math>(\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math> > TREE(G64)
|<math>(\lambda 1 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|95
|<math>> f_{\psi(\Omega_\omega)}\left(23\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<math>(\lambda 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|201
| > q(5)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/laver.lam JT & BF & 50_ft_lock]
|-||||||-
|331
| lim(BMS)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & JT & 50_ft_lock]
|-||||||-
|1850
|> Loader's number
|too large to show
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 JT]
|}

== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ1 by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BBλ(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ1 be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ1 up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!champion
!BBλ1
|-
|1
|
|0
|-
|2
|<math>1</math>
|1
|-
|3
|
|0
|-
|4
|<math>\lambda 1</math>
|4
|-
|5
|<math>\lambda 2</math>
|5
|-
|6
|<math>\lambda \lambda 1</math>
|6
|-
|7
|<math>\lambda \lambda 2</math>
|7
|-
|8
|<math>1 (\lambda 1)</math>
|<math>f(4) = 26</math>
|-
|9
|<math>\lambda \lambda 2</math>
|9
|-
|10
|<math>1 (\lambda \lambda 1)</math>
|<math>f(6) = 36</math>
|-
|11
|<math>1 (\lambda \lambda 2)</math>
|<math>f(7) = 41</math>
|-
|12
|<math>1 (1 (\lambda 1))</math>
|<math>f^{2}(4) = 266</math>
|-
|13
|<math>1 (\lambda \lambda 2)</math>
|<math>f(9) = 51</math>
|-
|14
|<math>1 (1 (\lambda \lambda 1))</math>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<math>1 (1 (\lambda \lambda 2))</math>
|<math>f^{2}(7) = f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<math>1 (1 (1 (\lambda 1)))</math>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<math>1 (1 (\lambda \lambda \lambda 2))</math>
|<math>f^2(9) = f(51)</math>
|-
|18
|<math>1 (\lambda 1) 1 (\lambda 1)</math>
|<math>f^4(4) </math>
|-
|19
|<math>1 (1 (1 (\lambda \lambda 2)))</math>
|<math>f^3(7)</math>
|-
|20
|<math>1 (\lambda \lambda 1) 1 (\lambda 1)</math>
|<math>f^6(4)</math>
|-
|21
|<math>1 (\lambda \lambda 2) 1 (\lambda 1)</math>
|<math>f^7(4)</math>
|-
|22
|<math>1 (1 (\lambda 1)) 1 (\lambda 1)</math>
|<math>f^{52}(4)</math>
|-
|...
|
|
|-
|28
|<math>1 (\lambda 1) 1 (\lambda 1) 1 (\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|-
|29
|<math>1(\lambda 1)(\lambda 1 2 1)(\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^{BB \lambda(f^4(4))+4}(4))+BB \lambda(f^4(4))+5}(4)</math>
|}
We can generalize BBλ1 to BBλα for ordinals α by using oracle function BBλα-1 for successor ordinal a, and oracle function (\n -> BBλα[n](n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== See Also ==
* [[Busy Beaver for SKI calculus|SKI calculus]]
* [[De Bruijn index]]
* https://oeis.org/A333479
* [https://www.mdpi.com/1099-4300/28/5/494 The Largest Number Representable in 64 Bits]. 26 Apr 2026. John Tromp.
* [https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d861 Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
* https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw
[[category:Functions]]

De Bruijn index

2026-05-09T23:01:00Z

ADucharme: creation of page by taking section from lambda calculus page

'''De Bruijn index''' is an alternative method to represent lambda calculus expressions.

Its attendant Busy Beaver problem is '''BBλ_db''' which uses De Bruijn index instead of binary to evaluate lambda calculus expression size. To calculate size, convert the lambda calculus expression into [[wikipedia:De_Bruijn_index|De Bruijn index]], then count the number of backslashes (lambdas) and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

For n < 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form, like BBλ(m) for m ≤ 20.
{| class="wikitable"
!BBλ_db(n)
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|12
|
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|mxdys and racheline
|-
|13
|
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|14
|
|<code>(\1 1 1) (\\1 (1 2) (\\2 (2 1)))</code>
|50_ft_lock
|-
|15
|
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|Gustavo Melo
|-
|18
|
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|50_ft_lock
|-
|22
|
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|23
|
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

Lambda Calculus

2026-05-09T22:49:45Z

ADucharme: removal of de bruijn index section, to be moved to new page

'''Lambda calculus''' is a model of computation developed by Alonzo Church (of Church-Turing thesis fame) in 1936. If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with [[wikipedia:Lambda_calculus|this article]].

'''BBλ''' is the [[Busy Beaver]] problem for lambda calculus, where BBλ(n) is the maximum normal form size of any closed lambda term of size n (or 0 if no closed term of size n exists). Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus], a binary prefix-free encoding for all closed lambda calculus terms. Pioneered by John Tromp, BBλ is uncomputable, and therefore grows faster than any computable function.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if any choice of beta-reductions will lead to this normal form and '''weakly normalizing''' if there exist divergent reduction paths which never reach the normal form.

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) = 0 for those values.
<math>C(n)</math> denotes Church numeral ''n'' = <math>\lambda f\lambda x. f^n(x)</math>.
In the last column, JT and BF abbreviate John Tromp and Bertram Felgenhauer. The [https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 smallest Cryptid known] currently is in 74 bits.

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
|4 || = 4 || <math>\lambda 1</math>
||| JT & BF
|-
|6 || = 6 || <math>\lambda\lambda 1</math>
||| JT & BF
|-
|7 || = 7 || <math>\lambda\lambda 2</math>
||| JT & BF
|-
|8 || = 8 || <math>\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|9 || = 9 || <math>\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|10 || = 10 || <math>\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|11 || = 11 || <math>\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|12 || = 12 || <math>\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|13 || = 13 || <math>\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|14 || = 14 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|15 || = 15 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|16 || = 16 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|17 || = 17 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|18 || = 18 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|19 || = 19 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|20 || = 20 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|21 || = 22 || <math>\lambda(\lambda 1 1) (1 (\lambda 2))</math>
| <math>\lambda(1(\lambda 2))(1(\lambda 2))</math>|| JT & BF
|-
|22 || = 24 || <math>\lambda(\lambda 1 1 1) (1 1)</math>
| <math>\lambda(1 1) (1 1) (1 1)</math>|| JT & BF
|-
|23 || = 26 || <math>\lambda(\lambda 1 1) (1 (\lambda\lambda 2))</math>
| <math>\lambda(1 (\lambda\lambda 2)) (1 (\lambda\lambda 2))</math>|| JT & BF
|-
|24 || = 30 || <math>\lambda(\lambda 1 1 1) (1 (\lambda 1))</math>
| <math>\lambda(1 (\lambda 1)) (1 (\lambda 1)) (1 (\lambda 1))</math>|| JT & BF
|-
|25 || = 42 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|26 || = 52 || <math>(\lambda 1 1) (\lambda\lambda 2 (1 2))</math>
| <math>\lambda\lambda 2 (\lambda\lambda 2 (1 2)) (1 (2 (\lambda\lambda 2 (1 2))))</math>|| JT & BF
|-
|27 || = 44 || <math>\lambda\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|28 || = 58 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 (\lambda 2))))</math>
| <math>\lambda 1 (\lambda\lambda 1 (3 (\lambda 2))) (1 (\lambda 2 (\lambda\lambda 1 (4 (\lambda 2)))))</math>|| JT & BF
|-
| 29 || = 223|| <math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 1)))</math>
| <math>\lambda B (B (1 B)) \text{ where}</math>
<math>B = (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (2 1))))</math>
||JT & BF
|-
|30
|= 160
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (1 2))</math>
|
<math>\lambda\lambda 2 B A (1 (2 B A)) \text{ where}</math>
<math>B = (\lambda\lambda 2 A (1 (2 A)))</math>,
<math>A = (\lambda\lambda 2 (1 2))</math>
|JT & BF
|-
|31
|= 267
|<math>(\lambda 1 1) (\lambda\lambda 2 (2 (1 2)))</math>
|
<math>\lambda\lambda 2 A (2 A (C (2 A))) \text{ where}</math>
<math>C = (2 A (2 A (1 B (2 A))))</math>,
<math>B = (\lambda 3 A (3 A (1 (3 A))))</math>,
<math>A = (\lambda\lambda 2 (2 (1 2)))</math>
|JT & BF
|-
|32
|= 298
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 (\lambda 2))))</math>
|
|JT & BF
|-
|33
|= 1812
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (1 (2 1))))</math>
|
<math>\lambda C (C (C (1 C))) \text{ where}</math>
<math>C = (B (B (B (1 B)))</math>,
<math>B = (A (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (1 (2 1)))))</math>
|JT & BF
|-
|34 || <math>= 327\,686</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^2}})</math>|| JT & BF
|-
|35 || <math>= 5 \cdot 3^{3^3} + 6</math>
<math> > 3.8 \times 10^{13}</math>
| <math>(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^3})</math>|| JT & BF
|-
|36 || <math>= 5 \cdot 2^{2^{2^3}} + 6</math>
<math> > 5.7 \times 10^{77}</math>
| <math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 2 (2 1))))</math>
| <math>C(2^{2^{2^3}})</math>|| JT & BF
|-
|37 || <math> = 2 + BB\lambda(35)</math>
| <math>\lambda(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>\lambda x. C(3^{3^3})</math>||mxdys & JT & dyuan & sligocki
|-
|38 || <math>= 5\cdot{2^{2^{2^{2^2}}}} + 6</math>
| <math>(\lambda 1 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^{2^2}}})</math>|| JT & BF & CppDS & mxdys & sligocki & dyuan & charles
|-
|39 || <math>\ge 10^{10^{12}}</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{3^3}})</math>|| JT & BF
|-
|40 || <math> > 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{15} 33</math>
|| mxdys & racheline
|-
|41 || <math>\ge 10^{10^{40}}</math>
| <math>(\lambda 1 (\lambda 1 1) 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge 2 + BB\lambda(40)</math>
|<math>\lambda(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math>,
<math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge 2 + BB\lambda(43)</math>
| <math>\lambda(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| ||
|-
|46 || <math> \ge 2 + BB\lambda(44)</math>
| <math>\lambda(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|47 || <math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
| <math>(\lambda 1 1 1)(\lambda\lambda 1 (1 2) (\lambda\lambda 2 (2 1)))</math>
| ||50_ft_lock
|-
|48 || <math> > 10 \uparrow\uparrow\uparrow\uparrow 4</math>
| <math>(\lambda 1 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math> > Graham's number
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 1 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right) )</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<math>(\lambda 1 1 1) (\lambda 1 (1 (\lambda\lambda\lambda 1 3 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right) )</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|<math>(\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math> > TREE(G64)
|<math>(\lambda 1 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|95
|<math>> f_{\psi(\Omega_\omega)}\left(23\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<math>(\lambda 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|201
| > q(5)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/laver.lam JT & BF & 50_ft_lock]
|-||||||-
|331
| lim(BMS)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & JT & 50_ft_lock]
|-||||||-
|1850
|> Loader's number
|too large to show
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 JT]
|}

== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ1 by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BBλ(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ1 be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ1 up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!champion
!BBλ1
|-
|1
|
|0
|-
|2
|<math>1</math>
|1
|-
|3
|
|0
|-
|4
|<math>\lambda 1</math>
|4
|-
|5
|<math>\lambda 2</math>
|5
|-
|6
|<math>\lambda \lambda 1</math>
|6
|-
|7
|<math>\lambda \lambda 2</math>
|7
|-
|8
|<math>1 (\lambda 1)</math>
|<math>f(4) = 26</math>
|-
|9
|<math>\lambda \lambda 2</math>
|9
|-
|10
|<math>1 (\lambda \lambda 1)</math>
|<math>f(6) = 36</math>
|-
|11
|<math>1 (\lambda \lambda 2)</math>
|<math>f(7) = 41</math>
|-
|12
|<math>1 (1 (\lambda 1))</math>
|<math>f^{2}(4) = 266</math>
|-
|13
|<math>1 (\lambda \lambda 2)</math>
|<math>f(9) = 51</math>
|-
|14
|<math>1 (1 (\lambda \lambda 1))</math>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<math>1 (1 (\lambda \lambda 2))</math>
|<math>f^{2}(7) = f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<math>1 (1 (1 (\lambda 1)))</math>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<math>1 (1 (\lambda \lambda \lambda 2))</math>
|<math>f^2(9) = f(51)</math>
|-
|18
|<math>1 (\lambda 1) 1 (\lambda 1)</math>
|<math>f^4(4) </math>
|-
|19
|<math>1 (1 (1 (\lambda \lambda 2)))</math>
|<math>f^3(7)</math>
|-
|20
|<math>1 (\lambda \lambda 1) 1 (\lambda 1)</math>
|<math>f^6(4)</math>
|-
|21
|<math>1 (\lambda \lambda 2) 1 (\lambda 1)</math>
|<math>f^7(4)</math>
|-
|22
|<math>1 (1 (\lambda 1)) 1 (\lambda 1)</math>
|<math>f^{52}(4)</math>
|-
|...
|
|
|-
|28
|<math>1 (\lambda 1) 1 (\lambda 1) 1 (\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|-
|29
|<math>1(\lambda 1)(\lambda 1 2 1)(\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^{BB \lambda(f^4(4))+4}(4))+BB \lambda(f^4(4))+5}(4)</math>
|}
We can generalize BBλ1 to BBλα for ordinals α by using oracle function BBλα-1 for successor ordinal a, and oracle function (\n -> BBλα[n](n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== See Also ==
* [[Busy Beaver for SKI calculus]]
* https://oeis.org/A333479
* [https://www.mdpi.com/1099-4300/28/5/494 The Largest Number Representable in 64 Bits]. 26 Apr 2026. John Tromp.
* [https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d861 Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
* https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw
[[category:Functions]]

Lambda Calculus

2026-05-09T22:47:24Z

ADucharme: introduction reorganization

'''Lambda calculus''' is a model of computation developed by Alonzo Church (of Church-Turing thesis fame) in 1936. If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with [[wikipedia:Lambda_calculus|this article]].

'''BBλ''' is the [[Busy Beaver]] problem for lambda calculus, where BBλ(n) is the maximum normal form size of any closed lambda term of size n (or 0 if no closed term of size n exists). Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus], a binary prefix-free encoding for all closed lambda calculus terms. Pioneered by John Tromp, BBλ is uncomputable, and therefore grows faster than any computable function.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if any choice of beta-reductions will lead to this normal form and '''weakly normalizing''' if there exist divergent reduction paths which never reach the normal form.

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) = 0 for those values.
<math>C(n)</math> denotes Church numeral ''n'' = <math>\lambda f\lambda x. f^n(x)</math>.
In the last column, JT and BF abbreviate John Tromp and Bertram Felgenhauer. The [https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 smallest Cryptid known] currently is in 74 bits.

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
|4 || = 4 || <math>\lambda 1</math>
||| JT & BF
|-
|6 || = 6 || <math>\lambda\lambda 1</math>
||| JT & BF
|-
|7 || = 7 || <math>\lambda\lambda 2</math>
||| JT & BF
|-
|8 || = 8 || <math>\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|9 || = 9 || <math>\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|10 || = 10 || <math>\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|11 || = 11 || <math>\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|12 || = 12 || <math>\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|13 || = 13 || <math>\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|14 || = 14 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|15 || = 15 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|16 || = 16 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|17 || = 17 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|18 || = 18 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|19 || = 19 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|20 || = 20 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|21 || = 22 || <math>\lambda(\lambda 1 1) (1 (\lambda 2))</math>
| <math>\lambda(1(\lambda 2))(1(\lambda 2))</math>|| JT & BF
|-
|22 || = 24 || <math>\lambda(\lambda 1 1 1) (1 1)</math>
| <math>\lambda(1 1) (1 1) (1 1)</math>|| JT & BF
|-
|23 || = 26 || <math>\lambda(\lambda 1 1) (1 (\lambda\lambda 2))</math>
| <math>\lambda(1 (\lambda\lambda 2)) (1 (\lambda\lambda 2))</math>|| JT & BF
|-
|24 || = 30 || <math>\lambda(\lambda 1 1 1) (1 (\lambda 1))</math>
| <math>\lambda(1 (\lambda 1)) (1 (\lambda 1)) (1 (\lambda 1))</math>|| JT & BF
|-
|25 || = 42 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|26 || = 52 || <math>(\lambda 1 1) (\lambda\lambda 2 (1 2))</math>
| <math>\lambda\lambda 2 (\lambda\lambda 2 (1 2)) (1 (2 (\lambda\lambda 2 (1 2))))</math>|| JT & BF
|-
|27 || = 44 || <math>\lambda\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|28 || = 58 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 (\lambda 2))))</math>
| <math>\lambda 1 (\lambda\lambda 1 (3 (\lambda 2))) (1 (\lambda 2 (\lambda\lambda 1 (4 (\lambda 2)))))</math>|| JT & BF
|-
| 29 || = 223|| <math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 1)))</math>
| <math>\lambda B (B (1 B)) \text{ where}</math>
<math>B = (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (2 1))))</math>
||JT & BF
|-
|30
|= 160
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (1 2))</math>
|
<math>\lambda\lambda 2 B A (1 (2 B A)) \text{ where}</math>
<math>B = (\lambda\lambda 2 A (1 (2 A)))</math>,
<math>A = (\lambda\lambda 2 (1 2))</math>
|JT & BF
|-
|31
|= 267
|<math>(\lambda 1 1) (\lambda\lambda 2 (2 (1 2)))</math>
|
<math>\lambda\lambda 2 A (2 A (C (2 A))) \text{ where}</math>
<math>C = (2 A (2 A (1 B (2 A))))</math>,
<math>B = (\lambda 3 A (3 A (1 (3 A))))</math>,
<math>A = (\lambda\lambda 2 (2 (1 2)))</math>
|JT & BF
|-
|32
|= 298
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 (\lambda 2))))</math>
|
|JT & BF
|-
|33
|= 1812
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (1 (2 1))))</math>
|
<math>\lambda C (C (C (1 C))) \text{ where}</math>
<math>C = (B (B (B (1 B)))</math>,
<math>B = (A (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (1 (2 1)))))</math>
|JT & BF
|-
|34 || <math>= 327\,686</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^2}})</math>|| JT & BF
|-
|35 || <math>= 5 \cdot 3^{3^3} + 6</math>
<math> > 3.8 \times 10^{13}</math>
| <math>(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^3})</math>|| JT & BF
|-
|36 || <math>= 5 \cdot 2^{2^{2^3}} + 6</math>
<math> > 5.7 \times 10^{77}</math>
| <math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 2 (2 1))))</math>
| <math>C(2^{2^{2^3}})</math>|| JT & BF
|-
|37 || <math> = 2 + BB\lambda(35)</math>
| <math>\lambda(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>\lambda x. C(3^{3^3})</math>||mxdys & JT & dyuan & sligocki
|-
|38 || <math>= 5\cdot{2^{2^{2^{2^2}}}} + 6</math>
| <math>(\lambda 1 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^{2^2}}})</math>|| JT & BF & CppDS & mxdys & sligocki & dyuan & charles
|-
|39 || <math>\ge 10^{10^{12}}</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{3^3}})</math>|| JT & BF
|-
|40 || <math> > 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{15} 33</math>
|| mxdys & racheline
|-
|41 || <math>\ge 10^{10^{40}}</math>
| <math>(\lambda 1 (\lambda 1 1) 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge 2 + BB\lambda(40)</math>
|<math>\lambda(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math>,
<math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge 2 + BB\lambda(43)</math>
| <math>\lambda(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| ||
|-
|46 || <math> \ge 2 + BB\lambda(44)</math>
| <math>\lambda(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|47 || <math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
| <math>(\lambda 1 1 1)(\lambda\lambda 1 (1 2) (\lambda\lambda 2 (2 1)))</math>
| ||50_ft_lock
|-
|48 || <math> > 10 \uparrow\uparrow\uparrow\uparrow 4</math>
| <math>(\lambda 1 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math> > Graham's number
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 1 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right) )</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<math>(\lambda 1 1 1) (\lambda 1 (1 (\lambda\lambda\lambda 1 3 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right) )</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|<math>(\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math> > TREE(G64)
|<math>(\lambda 1 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|95
|<math>> f_{\psi(\Omega_\omega)}\left(23\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<math>(\lambda 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|201
| > q(5)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/laver.lam JT & BF & 50_ft_lock]
|-||||||-
|331
| lim(BMS)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & JT & 50_ft_lock]
|-||||||-
|1850
|> Loader's number
|too large to show
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 JT]
|}

== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ1 by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BBλ(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ1 be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ1 up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!champion
!BBλ1
|-
|1
|
|0
|-
|2
|<math>1</math>
|1
|-
|3
|
|0
|-
|4
|<math>\lambda 1</math>
|4
|-
|5
|<math>\lambda 2</math>
|5
|-
|6
|<math>\lambda \lambda 1</math>
|6
|-
|7
|<math>\lambda \lambda 2</math>
|7
|-
|8
|<math>1 (\lambda 1)</math>
|<math>f(4) = 26</math>
|-
|9
|<math>\lambda \lambda 2</math>
|9
|-
|10
|<math>1 (\lambda \lambda 1)</math>
|<math>f(6) = 36</math>
|-
|11
|<math>1 (\lambda \lambda 2)</math>
|<math>f(7) = 41</math>
|-
|12
|<math>1 (1 (\lambda 1))</math>
|<math>f^{2}(4) = 266</math>
|-
|13
|<math>1 (\lambda \lambda 2)</math>
|<math>f(9) = 51</math>
|-
|14
|<math>1 (1 (\lambda \lambda 1))</math>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<math>1 (1 (\lambda \lambda 2))</math>
|<math>f^{2}(7) = f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<math>1 (1 (1 (\lambda 1)))</math>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<math>1 (1 (\lambda \lambda \lambda 2))</math>
|<math>f^2(9) = f(51)</math>
|-
|18
|<math>1 (\lambda 1) 1 (\lambda 1)</math>
|<math>f^4(4) </math>
|-
|19
|<math>1 (1 (1 (\lambda \lambda 2)))</math>
|<math>f^3(7)</math>
|-
|20
|<math>1 (\lambda \lambda 1) 1 (\lambda 1)</math>
|<math>f^6(4)</math>
|-
|21
|<math>1 (\lambda \lambda 2) 1 (\lambda 1)</math>
|<math>f^7(4)</math>
|-
|22
|<math>1 (1 (\lambda 1)) 1 (\lambda 1)</math>
|<math>f^{52}(4)</math>
|-
|...
|
|
|-
|28
|<math>1 (\lambda 1) 1 (\lambda 1) 1 (\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|-
|29
|<math>1(\lambda 1)(\lambda 1 2 1)(\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^{BB \lambda(f^4(4))+4}(4))+BB \lambda(f^4(4))+5}(4)</math>
|}
We can generalize BBλ1 to BBλα for ordinals α by using oracle function BBλα-1 for successor ordinal a, and oracle function (\n -> BBλα[n](n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== De Bruijn ==
We can use De Bruijn index instead of binary to evaluate lambda calculus size. To get the size of an expression, convert it into De Bruijn index then count the number of lambdas / backslashes and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

For n < 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form, like BBλ(m) for m ≤ 20.
{| class="wikitable"
!BBλ_db(n)
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|<math>\ge 3 \uparrow\uparrow 3 + 3 > 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|<math>\ge 3 \uparrow\uparrow 4 + 3 > 10^{10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|12
|<math>> 10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|14
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<code>(\1 1 1) (\\1 (1 2) (\\2 (2 1)))</code>
|50_ft_lock
|-
|15
|<math>> f_{\omega+1}(2 \uparrow\uparrow 6)</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|22
|<math>> f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

== See Also ==

* [[Busy Beaver for SKI calculus]]
* https://oeis.org/A333479
* [https://www.mdpi.com/1099-4300/28/5/494 The Largest Number Representable in 64 Bits]. 26 Apr 2026. John Tromp.
* [https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d861 Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
* https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw
[[category:Functions]]

Busy Beaver for lambda calculus

2026-05-09T22:31:42Z

ADucharme: ADucharme moved page Busy Beaver for lambda calculus to Lambda Calculus over redirect: standardization pf titles

#REDIRECT [[Lambda Calculus]]

Lambda Calculus

2026-05-09T22:31:42Z

ADucharme: ADucharme moved page Busy Beaver for lambda calculus to Lambda Calculus over redirect: standardization pf titles

'''Busy Beaver for lambda calculus''' ('''BBλ''') is a variation of the [[Busy Beaver]] problem for [https://en.wikipedia.org/wiki/Lambda_calculus lambda calculus] invented by John Tromp. BBλ(n) = the maximum normal form size of any closed lambda term of size n (or 0 if no closed term of size n exists). Like the traditional Busy Beaver functions, it is uncomputable (and in fact grows faster than any computable function). If you are not familiar with lambda calculus and beta-reduction, it is recommended to start with that article.

Size is measured in bits using [https://tromp.github.io/cl/Binary_lambda_calculus.html Binary Lambda Calculus] which is a binary prefix-free encoding for all closed lambda calculus terms.

== Analogy to Turing machines ==
We evaluate terms by applying ''beta-reductions'' until they reach a ''normal form''. As an analogy to [[Turing machines]]:
* ''Lambda terms'' are like TM configurations (tape + state + position).
* Applying ''beta-reduction'' to a term is like taking a TM step.
* A term is in ''normal form'' if no beta-reductions can be applied. This is like saying the term has halted.
* A term may or may not be reducible to a normal form. If it is, this is like saying the term halts.
* Determining whether a term is reducible to a normal form is an undecidable problem equivalent to the halting problem.

Note: That unlike for Turing machines, evaluating lambda terms is non-deterministic. Specifically, there may be multiple beta-reductions possible in a given term. However, if a term can be reduced to a normal form, that normal form is unique. It is not possible to reduce the original term to any different normal form. A term is '''strongly normalizing''' if any choice of beta-reductions will lead to this normal form and '''weakly normalizing''' if there exist divergent reduction paths which never reach the normal form.

== Proof of Uncomputability ==
The proof that BBλ(n) is uncomputable is very similar to Radó's original proof that Σ(n) is uncomputable. Proof by contradiction:

Assume BBλ is computable and so there exists a term ''f'' which computes it on [[wikipedia:Church_encoding|Church numerals]]. In other words: for all <math>n \in \N</math>: <math>(f \; C_n)</math> beta reduces to normal form <math>C_{BB\lambda(n)}</math> (where <math>C_n</math> denotes the Church numeral ''n''). Denote the binary lambda encoded size of ''f'' as ''k''. Consider the term <math>f \; (C_2 \; C_n)</math> which has size <math>2+k+2+(5\times2+6)+(5n+6) = 5n + k + 26</math> bits. This term reduces to <math>C_{BB\lambda(n^2)}</math> which has size <math>5 \cdot BB\lambda(n^2) + 6</math> bits. But for sufficiently large n, <math>n^2 > 5n + k + 26</math> and so <math>5 \cdot BB\lambda(n^2) + 6 > BB\lambda(5n + k + 26)</math>. But this is a contradiction, we've found a <math>5n + k + 26</math> bit term which reduces to a normal form larger than <math>BB\lambda(5n + k + 26)</math>.

Thus BBλ(n) is uncomputable. A variation of this argument shows that BBλ(n) eventually dominates all computable functions.

== Binary Lambda Encoding ==
A lambda term using [https://en.wikipedia.org/wiki/De_Bruijn_indices De Bruijn indexes] is defined inductively as:
* Variables: For any <math>n \in \mathbb{Z}^+</math>, Var(''n'') is a term. It represents a variable bound by the lambda expression ''n'' above this one (the De Bruijn index). It is typically written simply as <code>n</code>.
* Lambdas: For any term ''T'', Lam(''T'') is a term. It represents a unary function with function body ''T''. It is typically written <math>\lambda T</math> or <code>\T</code>.
* Applications: For any terms ''T, U'', App(''T, U'') is a term. It represents applying function ''T'' to argument ''U''. It is typically written <code>(T U)</code>.

We can think of this as a tree where each variable is a leaf, a lambda is a node with one child and applications are nodes with 2 children. A term is '''closed''' if every variable is bound. In other words, for every Var(''n'') leaf node, there exists ''n'' Lam() nodes above it in the tree of the term.

Encoding (''blc()'') is defined recursively:
<math display="block">\begin{array}{l}
blc(Var(n)) & = & 1^n 0 \\
blc(Lam(T)) & = & 00 \; blc(T) \\
blc(App(T, U)) & = & 01 \; blc(T) \; blc(U) \\
\end{array}</math>

For example, the [https://en.wikipedia.org/wiki/Church_encoding#Church_numerals Church numeral] 2: <math>\lambda f x. (f \; (f \; x))</math> = <code>\\(2 (2 1))</code> = <code>Lam(Lam(App(Var(2), App(Var(2), Var(1))))</code> is encoded as <code>00 00 01 110 01 110 10</code> or simply <code>0000011100111010</code> (spaces are not part of the encoding, only used for demonstration purposes) and thus has size 16 bits.

== Text Encoding conventions ==
For human readability, a text encoding and set of conventions is used in this article. As described earlier we encode a lambda term as:
* Var(''n'') -> <code>n</code>
* Lam(''T'') -> <code>(\T)</code>
* App(''T, U'') -> <code>(T U)</code>

However, parentheses are also dropped in certain cases by convention:
* The outermost parentheses are dropped: <code>Lam(1)</code> -> <code>\1</code> and <code>App(1, 2)</code> -> <code>1 2</code>.
* Parentheses are dropped immediately inside a Lam: <code>Lam(Lam(1))</code> -> <code>\\1</code> and <code>Lam(App(1, 1))</code> -> <code>\1 1</code>.
* Parentheses are dropped in nested Apps using left associativity: <code>App(App(1, 2), 3)</code> -> <code>1 2 3</code>. (Note: parentheses are still required for <code>App(1, App(2, 3))</code> -> <code>1 (2 3)</code>).

This is the convention used in John Tromp's code and so is used here for consistency.

== Champions ==
There are no closed lambda terms of size 0, 1, 2, 3 or 5 and so BBλ(n) = 0 for those values.
<math>C(n)</math> denotes Church numeral ''n'' = <math>\lambda f\lambda x. f^n(x)</math>.
In the last column, JT and BF abbreviate John Tromp and Bertram Felgenhauer. The [https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 smallest Cryptid known] currently is in 74 bits.

{| class="wikitable"
|+
!n
!BBλ(n)
!Champion
!Normal form
!Discovered By
|-
|4 || = 4 || <math>\lambda 1</math>
||| JT & BF
|-
|6 || = 6 || <math>\lambda\lambda 1</math>
||| JT & BF
|-
|7 || = 7 || <math>\lambda\lambda 2</math>
||| JT & BF
|-
|8 || = 8 || <math>\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|9 || = 9 || <math>\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|10 || = 10 || <math>\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|11 || = 11 || <math>\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|12 || = 12 || <math>\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|13 || = 13 || <math>\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|14 || = 14 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|15 || = 15 || <math>\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|16 || = 16 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|17 || = 17 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|18 || = 18 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|19 || = 19 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 2</math>
||| JT & BF
|-
|20 || = 20 || <math>\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda\lambda 1</math>
||| JT & BF
|-
|21 || = 22 || <math>\lambda(\lambda 1 1) (1 (\lambda 2))</math>
| <math>\lambda(1(\lambda 2))(1(\lambda 2))</math>|| JT & BF
|-
|22 || = 24 || <math>\lambda(\lambda 1 1 1) (1 1)</math>
| <math>\lambda(1 1) (1 1) (1 1)</math>|| JT & BF
|-
|23 || = 26 || <math>\lambda(\lambda 1 1) (1 (\lambda\lambda 2))</math>
| <math>\lambda(1 (\lambda\lambda 2)) (1 (\lambda\lambda 2))</math>|| JT & BF
|-
|24 || = 30 || <math>\lambda(\lambda 1 1 1) (1 (\lambda 1))</math>
| <math>\lambda(1 (\lambda 1)) (1 (\lambda 1)) (1 (\lambda 1))</math>|| JT & BF
|-
|25 || = 42 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|26 || = 52 || <math>(\lambda 1 1) (\lambda\lambda 2 (1 2))</math>
| <math>\lambda\lambda 2 (\lambda\lambda 2 (1 2)) (1 (2 (\lambda\lambda 2 (1 2))))</math>|| JT & BF
|-
|27 || = 44 || <math>\lambda\lambda(\lambda 1 1) (\lambda 1 (2 1))</math>
| <math>\lambda\lambda 1 (\lambda 1 (2 1)) (1 (1 (\lambda 1 (2 1))))</math>|| JT & BF
|-
|28 || = 58 || <math>\lambda(\lambda 1 1) (\lambda 1 (2 (\lambda 2))))</math>
| <math>\lambda 1 (\lambda\lambda 1 (3 (\lambda 2))) (1 (\lambda 2 (\lambda\lambda 1 (4 (\lambda 2)))))</math>|| JT & BF
|-
| 29 || = 223|| <math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 1)))</math>
| <math>\lambda B (B (1 B)) \text{ where}</math>
<math>B = (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (2 1))))</math>
||JT & BF
|-
|30
|= 160
|<math>(\lambda 1 1 1) (\lambda\lambda 2 (1 2))</math>
|
<math>\lambda\lambda 2 B A (1 (2 B A)) \text{ where}</math>
<math>B = (\lambda\lambda 2 A (1 (2 A)))</math>,
<math>A = (\lambda\lambda 2 (1 2))</math>
|JT & BF
|-
|31
|= 267
|<math>(\lambda 1 1) (\lambda\lambda 2 (2 (1 2)))</math>
|
<math>\lambda\lambda 2 A (2 A (C (2 A))) \text{ where}</math>
<math>C = (2 A (2 A (1 B (2 A))))</math>,
<math>B = (\lambda 3 A (3 A (1 (3 A))))</math>,
<math>A = (\lambda\lambda 2 (2 (1 2)))</math>
|JT & BF
|-
|32
|= 298
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (2 (\lambda 2))))</math>
|
|JT & BF
|-
|33
|= 1812
|<math>\lambda(\lambda 1 1) (\lambda 1 (1 (1 (2 1))))</math>
|
<math>\lambda C (C (C (1 C))) \text{ where}</math>
<math>C = (B (B (B (1 B)))</math>,
<math>B = (A (A (A (1 A)))</math>,
<math>A = (1 (\lambda 1 (1 (1 (2 1)))))</math>
|JT & BF
|-
|34 || <math>= 327\,686</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^2}})</math>|| JT & BF
|-
|35 || <math>= 5 \cdot 3^{3^3} + 6</math>
<math> > 3.8 \times 10^{13}</math>
| <math>(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^3})</math>|| JT & BF
|-
|36 || <math>= 5 \cdot 2^{2^{2^3}} + 6</math>
<math> > 5.7 \times 10^{77}</math>
| <math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 2 (2 1))))</math>
| <math>C(2^{2^{2^3}})</math>|| JT & BF
|-
|37 || <math> = 2 + BB\lambda(35)</math>
| <math>\lambda(\lambda 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>\lambda x. C(3^{3^3})</math>||mxdys & JT & dyuan & sligocki
|-
|38 || <math>= 5\cdot{2^{2^{2^{2^2}}}} + 6</math>
| <math>(\lambda 1 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
| <math>C(2^{2^{2^{2^2}}})</math>|| JT & BF & CppDS & mxdys & sligocki & dyuan & charles
|-
|39 || <math>\ge 10^{10^{12}}</math>
| <math>(\lambda 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{3^3}})</math>|| JT & BF
|-
|40 || <math> > 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{15} 33</math>
|| mxdys & racheline
|-
|41 || <math>\ge 10^{10^{40}}</math>
| <math>(\lambda 1 (\lambda 1 1) 1) (\lambda\lambda 2 (2 (2 1)))</math>
| <math>C(3^{3^{85}})</math>||mxdys
|-
|42 ||<math> \ge 2 + BB\lambda(40)</math>
|<math>\lambda(\lambda 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|43 ||<math> > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
|<math>(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;(\lambda y.y\;C(2)\;T(n))</math>,
<math>k > 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow\uparrow 2 \uparrow\uparrow 8</math>
||mxdys
|-
|44 || <math> > 10 \uparrow\uparrow\uparrow 10 \uparrow\uparrow\uparrow 16</math>
| <math>(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)\text{ where}</math>
<math>T(0)=x</math>,
<math>T(n+1)=T(n)\;C(2)\;T(n)</math>,
<math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33</math>||
|-
|45 || <math> \ge 2 + BB\lambda(43)</math>
| <math>\lambda(\lambda 1 1) (\lambda 1 (\lambda 1 (\lambda\lambda 2 (2 1)) 2))</math>
| ||
|-
|46 || <math> \ge 2 + BB\lambda(44)</math>
| <math>\lambda(\lambda 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| ||
|-
|47 || <math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
| <math>(\lambda 1 1 1)(\lambda\lambda 1 (1 2) (\lambda\lambda 2 (2 1)))</math>
| ||50_ft_lock
|-
|48 || <math> > 10 \uparrow\uparrow\uparrow\uparrow 4</math>
| <math>(\lambda 1 1 1 1 1) (\lambda 1 (\lambda\lambda 2 (2 1)) 1)</math>
| <math>\lambda x.T(k)</math> where <math>T(0)=x,\;T(n+1)=T(n)\;C(2)\;T(n)</math> and <math>k > (2\uparrow\uparrow)^{(2\uparrow\uparrow)^{(2\uparrow\uparrow)^{15} 33 - 1} 33 - 1} 33</math>||
|-
|49
|<math>> f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right)</math> > Graham's number
|<math>(\lambda 1 1) (\lambda 1 (1 (\lambda\lambda 1 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega+1}\left(\frac{2 \uparrow\uparrow 6}{2}\right) )</math>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|...
|
|
|
|
|-
|61
|<math>> f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right)</math>
|<math>(\lambda 1 1 1) (\lambda 1 (1 (\lambda\lambda\lambda 1 3 2 (\lambda\lambda 2 (2 1)))))</math>
|<math>C(f_{\omega^{2 \uparrow\uparrow 18-1}}\left(2\right) )</math>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|...
|
|
|
|
|-
|86
|<math>> f_{\omega^{\omega^{2}}}\left(2\right)</math>
|<math>(\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|90
|<math>> f_{\zeta_0}\left(15\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|...
|
|
|
|
|-
|94
|<math>> f_{\psi(\Omega_\omega)}\left(12\right)</math> > TREE(G64)
|<math>(\lambda 1 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|95
|<math>> f_{\psi(\Omega_\omega)}\left(23\right)</math>
|<math>(\lambda 1 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) (\lambda\lambda 2 (2 (2 1)))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-
|96
|<math>> f_{\psi(\Omega_\omega)}\left(f_{\omega^{\omega^{2}}}\left(2\right)\right)</math>
|<math>(\lambda 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|100
|<math>> f_{\psi(\Omega_\omega)+1}\left(4\right)</math>
|<math>(\lambda 1 1 (\lambda 1 (\lambda\lambda\lambda\lambda 1 4 4 4 3 2 1) 1 1 1 1) 1) (\lambda\lambda 2 (2 1))</math>
|
|[https://docs.google.com/document/d/1xlzaEQGarqnCocf4R2UWfqE3ck8YF_P32CmYxGXLhAI/edit?tab=t.0 Patcail]
|-||||||-
|201
| > q(5)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/laver.lam JT & BF & 50_ft_lock]
|-||||||-
|331
| lim(BMS)
|too large to show
|
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam Patcail & JT & 50_ft_lock]
|-||||||-
|1850
|> Loader's number
|too large to show
|
|[https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634 JT]
|}

== Oracle Busy Beaver ==
While BBλ grows uncomputably fast, one can define functions that grow much faster.

Let's define a higher order busy beaver function BBλ1 by providing oracle access to BBλ.

This is done by enriching the set of terms and possible reduction steps considered in the BB definition.

A 1-closed term is a term in de Bruijn notation that is closed with 1 additional lambda in front. Any variable bound to that lambda is a free variable '''f''' in the term.

An oracle reduction step reduces '''f''' t, where t is a closed normal form of size s, to Church numeral BBλ(s).

Note that this is almost identical to the oracle steps in Barendregt and Klop's "Applications of infinitary lambda calculus", except that they require t itself to be a church numeral. Allowing arbitrary closed t makes oracle steps more widely applicable while aligning with BBλ's focus on term sizes.

Now let BBλ1 be the maximum beta/oracle normal form size of any 1-closed lambda term of size n, or 0 if no 1-closed term of size n exists. This appears as sequence [[oeis:A385712|A385712]] in the OEIS.

The following table shows values of BBλ1 up to 22 plus a lower bound for 28, with larger values expressed in terms of function <math>f(n) = 6 + 5 \times BB \lambda(n)</math>:
{| class="wikitable"
|+
!n
!champion
!BBλ1
|-
|1
|
|0
|-
|2
|<math>1</math>
|1
|-
|3
|
|0
|-
|4
|<math>\lambda 1</math>
|4
|-
|5
|<math>\lambda 2</math>
|5
|-
|6
|<math>\lambda \lambda 1</math>
|6
|-
|7
|<math>\lambda \lambda 2</math>
|7
|-
|8
|<math>1 (\lambda 1)</math>
|<math>f(4) = 26</math>
|-
|9
|<math>\lambda \lambda 2</math>
|9
|-
|10
|<math>1 (\lambda \lambda 1)</math>
|<math>f(6) = 36</math>
|-
|11
|<math>1 (\lambda \lambda 2)</math>
|<math>f(7) = 41</math>
|-
|12
|<math>1 (1 (\lambda 1))</math>
|<math>f^{2}(4) = 266</math>
|-
|13
|<math>1 (\lambda \lambda 2)</math>
|<math>f(9) = 51</math>
|-
|14
|<math>1 (1 (\lambda \lambda 1))</math>
|<math>f^{2}(6) = f(36) = 25 \times 2^{2^{2^{3}}}+36 > 2.85 \times 10^{78}</math>
|-
|15
|<math>1 (1 (\lambda \lambda 2))</math>
|<math>f^{2}(7) = f(41) \geq 25 \times 3^{3^{85}}+36 > 10^{10^{40}}</math>
|-
|16
|<math>1 (1 (1 (\lambda 1)))</math>
|<math>f^{3}(4) = f(266)</math>
|-
|17
|<math>1 (1 (\lambda \lambda \lambda 2))</math>
|<math>f^2(9) = f(51)</math>
|-
|18
|<math>1 (\lambda 1) 1 (\lambda 1)</math>
|<math>f^4(4) </math>
|-
|19
|<math>1 (1 (1 (\lambda \lambda 2)))</math>
|<math>f^3(7)</math>
|-
|20
|<math>1 (\lambda \lambda 1) 1 (\lambda 1)</math>
|<math>f^6(4)</math>
|-
|21
|<math>1 (\lambda \lambda 2) 1 (\lambda 1)</math>
|<math>f^7(4)</math>
|-
|22
|<math>1 (1 (\lambda 1)) 1 (\lambda 1)</math>
|<math>f^{52}(4)</math>
|-
|...
|
|
|-
|28
|<math>1 (\lambda 1) 1 (\lambda 1) 1 (\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^3(4))}(4)</math>
|-
|29
|<math>1(\lambda 1)(\lambda 1 2 1)(\lambda 1)</math>
|<math>\ge f^{BB \lambda(f^{BB \lambda(f^4(4))+4}(4))+BB \lambda(f^4(4))+5}(4)</math>
|}
We can generalize BBλ1 to BBλα for ordinals α by using oracle function BBλα-1 for successor ordinal a, and oracle function (\n -> BBλα[n](n)) for limit ordinal α, assuming well-defined fundamental sequences up to α. Because of limited oracle inputs, all oracle busy beavers have identical values up to n=11.

== De Bruijn ==
We can use De Bruijn index instead of binary to evaluate lambda calculus size. To get the size of an expression, convert it into De Bruijn index then count the number of lambdas / backslashes and numbers. By example, <code>(\1 1) (\\2 (1 2))</code> is size 8 because it has 3 backslashes and 5 numbers.

For n < 7, BBλ_db(n) = n is trivial and can be achieved via picking any size n term already in normal form, like BBλ(m) for m ≤ 20.
{| class="wikitable"
!BBλ_db(n)
!Value
!Champion
!Discovered By
|-
|7
|≥ 7
|<code>\1 1 1 1 1 1</code>
|
|-
|8
|≥ 16
|<code>(\1 1) (\\2 (1 2))</code>
|[[User:Azerty|Azerty]] & John Tromp & Bertram Felgenhauer
|-
|9
|≥ 68
|<code>(\1 1) (\\2 (2 (1 2)))</code>
|John Tromp & Bertram Felgenhauer
|-
|10
|<math>\ge 3 \uparrow\uparrow 3 + 3 > 7.625 \times 10^{12}</math>
|<code>(\1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|11
|<math>\ge 3 \uparrow\uparrow 4 + 3 > 10^{10^{12}}</math>
|<code>(\1 1 1 1) (\\2 (2 (2 1)))</code>
|
|-
|12
|<math>> 10 {\uparrow}^{3} 16</math>
|<code>(\1 1 1) (\1 (\\2 (2 1)) 1)</code>
|mxdys and racheline
|-
|13
|<math>> 10 {\uparrow}^{3} 10 {\uparrow}^{3} 10 {\uparrow}^{2} 6</math>
|<code>(\1 1) (\1 (\1 (\\2 (2 1)) 2))</code>
|mxdys
|-
|14
|<math>> f_{\omega}\left(f_{5}\left(2\right)\right)</math>
|<code>(\1 1 1) (\\1 (1 2) (\\2 (2 1)))</code>
|50_ft_lock
|-
|15
|<math>> f_{\omega+1}(2 \uparrow\uparrow 6)</math>
|<code>(\1 1) (\1 (1 (\\1 2 (\\2 (2 1)))))</code>
|[https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/melo.lam Gustavo Melo]
|-
|18
|<math>> f_{\omega^\omega}(2 \uparrow\uparrow 18)</math>
|<code>(\1 1 1) (\1 (1 (\\\1 3 2 (\\2 (2 1)))))</code>
|[https://tromp.github.io/blog/2026/01/28/largest-number-revised 50_ft_lock]
|-
|22
|<math>> f_{\omega^{\omega+2}}(2)</math>
|<code>(\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|23
|<math>> f_{\zeta_0}(15)</math>
|<code>(\1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|24
|<math>> f_{\psi(\Omega_\omega)}(12)</math>
|<code>(\1 1 1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) (\\2 (2 1))</code>
|Patcail
|-
|25
|<math>> f_{\psi(\Omega_\omega)}(f_{\omega^{\omega+2}}(2))</math>
|<code>(\1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|-
|26
|<math>> f_{\psi(\Omega_\omega+1)}(4)</math>
|<code>(\1 1 (\1 (\\\\1 4 4 4 3 2 1) 1 1 1 1) 1) (\\2 (2 1))</code>
|Patcail
|}

== See Also ==

* [[Busy Beaver for SKI calculus]]
* https://oeis.org/A333479
* [https://www.mdpi.com/1099-4300/28/5/494 The Largest Number Representable in 64 Bits]. 26 Apr 2026. John Tromp.
* [https://gist.github.com/tromp/86b3184f852f65bfb814e3ab0987d861 Binary Lambda Calculus]. John Tromp.
* https://github.com/tromp/AIT/tree/master/BB
* https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw
[[category:Functions]]

BB(4,3)

2026-05-09T19:59:23Z

ADucharme: /* Stage 3 */ new inductive decider row

The Busy Beaver problem for 4 states and 3 symbols is unsolved. The existence of [[Cryptids]] in the domain is given by the discovery of [[Bigfoot]] in [[BB(3,3)]]. The current [[Champions#3-Symbol TMs|champion]] is {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} which was discovered by Pavel Kropitz in May 2024 along with 6 other long running machines. It was [[User:Polygon/Page for analyses#1RB1RD1LC 2LB1RB1LC 1RZ1LA1LD 0RB2RA2RD (bbch)|analyzed by Polygon]] in Oct 2025, demonstrating the lower bounds:

<math display="block">S(4,3) > \Sigma(4,3) > 10 \uparrow^{4} 4</math>

== Top Halters ==
The longest running halting BB(4,3) TMs are split amongst two classes: the pentational and hexational TMs found by Pavel Kropitz outlined in the Potential Champions section, and the tetrational TMs found by comprehensive holdout filtering by Terry Ligocki. 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 longest running halters found by Pavel Kropitz are:
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|}
The top 20 scoring halting machines found by comprehensive search are:
{| class="wikitable"
|+
!Standard format
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|~10 ↑↑ 1023.47221
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 619.07737
|Andrew Ducharme
|-
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|~10 ↑↑ 512.10945
|Andrew Ducharme
|-
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|~10 ↑↑ 439.02781
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 234.06408
|Andrew Ducharme
|-
|{{TM|1RB0LC1RC_1LA2RB1LB_1RC2LA0RD_2LB1RZ2LC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LA1RA_1LA0RC1LC_1LC2RB0LD_2RA1RZ2RC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_2RC1LB2RD_1LD2LA0LB_0LA1RZ0LC|halt}}
|~10 ↑↑ 178.48320
|Andrew Ducharme
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA1LB_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB1LD0RC_2LC0RB1RA_1RA0LB1RD_0LA2LA1RZ|halt}}
|~10 ↑↑ 158.81916
|Andrew Ducharme
|-
|{{TM|1RB1RC1RB_1LC0RA2LD_2RA0LD1RZ_0LB2LD1RD|halt}}
|~10 ↑↑ 154.52968
|Andrew Ducharme
|-
|{{TM|1RB1LA1RD_2LA0LC2LD_1RZ2RA2LB_0LC2RC1RA|halt}}
|~10 ↑↑ 147.26175
|Andrew Ducharme
|-
|{{TM|1RB0RB1LC_2LC0LD1RA_2RB2LD1RZ_2LA2LB0LD|halt}}
|~10 ↑↑ 141.44248
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ2LD1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC1LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA---|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|}

== Potential Champions ==
In May 2024, [https://discord.com/channels/960643023006490684/1026577255754903572/1243253180297646120 Pavel Kropitz found 7 halting TMs] that run for a large number of steps. Four of these are equivalent and were [https://discord.com/channels/960643023006490684/1331570843829932063/1337228898068463718 analyzed by Racheline] in February 2025, while the remaining three were [[User:Polygon/Page for analyses|analyzed by Polygon in October 2025.]]
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}*
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}
|<math>10 \uparrow\uparrow 9.873987</math>
|}
<nowiki>*</nowiki>equivalent to {{TM|0RB1RZ1RC_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}, {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ0RA|halt}} and {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ1RB|halt}}.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in December 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~633B(illion) TMs of which ~34.4B TMs were holdouts. Also found were ~206B halting TMs and ~392B infinite TMs. The number of holdouts was reduced to ~461M TMs (a 99.93% reduction).

Two C++ programs were run before the filters in the table.
<pre>
lr_enum 4 3 8 /dev/null /dev/null 4x3.unk.txt false
00 <= XX < 47: lr_enum_continue 4x3.in.XX 1000 /dev/null /dev/null 4x3.unk.txt.XX XX false
</pre>
Both do the initial enumeration and simple filtering. The "/dev/null" in both commands would be files where the halting and infinite TMs would be stored. The first command generates the TMs from a [[TNF]] tree for BB(4,3) of depth 8 and outputs the holdouts to 4x3.unk.txt. This file was then divided into 48 pieces, 4x3.in.XX, 0 <= XX < 47. The second commands (one for each XX) continues the enumeration by running each TM for 1,000 steps. It classifies each as halting, infinite, or unknown/holdout. Again, the halting and infinite TMs are "written" to /dev/null, i.e., they aren't saved. The holdouts are stored in 48 files: 4x3.unk.txt.XX.

For these runs the first command generated a total of ~45M TMs: ~1.86M halting, ~774K infinite, and ~42.0M holdouts. The second took the ~42.0M holdout TMs and generated a total of ~633B TMs: ~206B halting, ~392B infinite, and ~34.4B holdouts. These holdouts were used as a starting point of the filters below.

The "Description" column in the table below contain the command run. Two options are not given, "--infile=..." and an "--outfile=...". These are necessary and specify where to read and write the results, respectively. Note: The work flow was to divide the input holdouts into 48 pieces, run the command on each piece simultaneously on one of 48 cores, and then combine the 48 results into a group of holdouts.

The details are given in this table:

(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"
!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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|34,413,860,527
|30,874,934,791
|10.28%
|646.6
|1,520.36
|14,784.57
|style="text-align:left" |Reverse_Engineer_Filter.py
|rowspan="10" style="text-align:left" |[https://drive.google.com/drive/folders/1KMOVgngtUVMEA7EjxtNcsgksQ5Y4tby9?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|30,874,934,791
|12,942,386,396
|58.08%
|4,134.8
|1,204.72
|2,074.19
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|12,942,386,396
|4,534,322,415
|64.97%
|3,361.1
|694.88
|1,069.62
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|4,534,322,415
|2,959,598,830
|34.73%
|3,318.1
|131.83
|379.59
|style="text-align:left" |CPS_Filter.py --block-size=3
|-
|style="text-align:left" |Terry Ligocki
|2,959,598,830
|1,651,940,618
|44.18%
|2,700.6
|134.50
|304.42
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --no-steps --time=0.002 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|1,651,940,618
|854,984,279
|48.24%
|2,276.3
|97.25
|201.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.005 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|854,984,279
|683,163,325
|20.10%
|430.1
|110.96
|552.15
|style="text-align:left" |CPS_Filter.py --block-size=4 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|683,163,325
|460,916,384
|32.53%
|5,507.9
|11.21
|34.45
|style="text-align:left" |CPS_Filter.py --min-block-size=1 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:center" |'''Cumulative'''
|'''632,656,365,801'''
|'''460,916,384'''
|'''99.93%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

== Phase 2 ==

When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===

Starting from the results of Phase 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 33 variations were run. The holdouts were reduced from ~461B TMs to ~33.9M TMs (a 92.7% reduction). The details are given in the table below, including links to the Google Drive with the holdouts. Entries with multiple lines represent runs where all the commands in the "Description" were applied during one run.

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|460,916,384
|234,834,703
|49.05%
|96.7
|649.48
|1,324.10
|style="text-align:left" | chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 1000 NG_n 2 run
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1tFtg1eFC-AdqCzh7XNmx5O2mTQwtaNbm?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|234,834,703
|160,518,206
|31.65%
|70.9
|291.33
|920.57
|style="text-align:left" | chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 1000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|160,518,206
|132,296,033
|17.58%
|41.5
|188.86
|1,074.17
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 6 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|132,296,033
|113,193,595
|14.44%
|54.9
|96.57
|668.77
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|113,193,595
|85,920,795
|24.09%
|106.8
|70.96
|294.52
|style="text-align:left" | chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|85,920,795
|78,674,774
|8.43%
|28.9
|69.62
|825.51
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|78,674,774
|73,228,547
|6.92%
|68.7
|22.02
|318.04
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|73,228,547
|67,014,897
|8.49%
|23.2
|74.50
|878.02
|style="text-align:left" | chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|67,014,897
|57,625,231
|14.01%
|75.6
|34.49
|246.13
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|57,625,231
|48,070,606
|16.58%
|645.4
|4.11
|24.80
|style="text-align:left" | chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 30000 NG_n 10 run
|-
|style="text-align:left" |Terry Ligocki
|48,070,606
|44,254,286
|7.94%
|166.3
|6.38
|80.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 6 H 1 tH 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|44,254,286
|40,836,159
|7.72%
|188.3
|5.04
|65.29
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|40,836,159
|37,460,692
|8.27%
|192.3
|4.88
|58.99
|style="text-align:left" |
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 2 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 3 n 2 run 
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 3000 NG_n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 4 H 1 tH 1 n 2 run 
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 3 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 2 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|37,460,692
|36,167,570
|3.45%
|237.7
|1.51
|43.77
|style="text-align:left" |
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 3 H 0 tH 1 n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 10 chr_H 8 MitM_CTL NG maxT 10000 NG_n 4 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 2 run 
|-
|style="text-align:left" |Terry Ligocki
|36,167,570
|34,642,544
|4.22%
|467.2
|0.91
|21.50
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,642,544
|34,339,943
|0.87%
|383.1
|0.22
|25.12
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 0 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,339,943
|33,860,069
|1.40%
|666.5
|0.20
|14.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 2 tH 2 n 8 run
|-
|style="text-align:center" |'''Cumulative'''
|'''460,916,384'''
|'''33,860,069'''
|'''92.70%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

=== Stage 2 ===

Starting from the results of Stage 1, Terry Ligocki ran a variety of enumeration and decider codes. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~33.9M TMs to ~9.4M TMs (a 72.2% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|33,860,069
|21,065,769
|37.79%
|93.0
|38.20
|101.11
|style="text-align:left" |lr_enum_continue 4x3.in.txt 1000000 4x3.halt.txt 4x3.inf.txt 4x3.holdouts.txt 00 false
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|21,065,769
|18,949,009
|10.05%
|5,566.1
|0.11
|1.05
|style="text-align:left" |Enumerate.py max-loops 100_000 block-size 2 --tape-limit 1_000 --no-steps --time 1.0 --recursive --exp-linear-rules --lin-steps 0 --no-ctl --no-reverse-engineer --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,949,009
|18,138,027
|4.28%
|0.4
|511.59
|11953.46
|style="text-align:left" |Reverse_Engineer_Filter.py --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,138,027
|11,985,999
|33.92%
|4.8
|352.73
|1,039.95
|style="text-align:left" | chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|11,985,999
|9,988,715
|16.66%
|640.4
|0.87
|5.20
|style="text-align:left" |
chr_LRUH 24 chr_H 16 MitM_CTL NG maxT 30000 NG_n 3 run 
chr_LRUH 14 chr_H 2 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 5 run 
chr_asth 0 chr_LRUH 48 chr_H 48 MitM_CTL NG maxT 30000 NG_n 5 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 3 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 1 n 4 run 
chr_LRUH 14 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 0 n 6 run 
chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 18 chr_H 16 MitM_CTL NG maxT 30000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 1 tH 0 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 3 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|9,988,715
|9,401,447
|5.88%
|1,398.7
|0.12
|1.98
|style="text-align:left" |
chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 100000 NG_n 5 run 
chr_LRUH 22 chr_H 12 MitM_CTL NG maxT 100000 NG_n 6 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 100000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 16 H 1 tH 0 n 10 run 
chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 1000000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 6 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 2 tH 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 6 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 1 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 5 n 2 run 
|-
|style="text-align:center" |'''Cumulative'''
|'''33,860,069'''
|'''9,401,447'''
|'''72.23%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

==== Stage 3 ====
Starting from the results of Stage 2, Andrew Ducharme ran a variety of Ligocki and @mxdys deciders. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~9.4M TMs to ~5.6M (a 40.0% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:
{| 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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
| style="text-align:left" |Andrew Ducharme
|9,401,447
|7,753,702
|17.53%
|988.7
|
|
| style="text-align:left" |Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000 --block-mult=3 --time=0.5 --lin-steps=0 --no-ctl
| rowspan="29" style="text-align:left" |[https://drive.google.com/drive/folders/11FYe0CVDPczcgt4put3vsdGbeMWkM1S1?usp=sharing Google Drive]
|-
| style="text-align:left" |Andrew Ducharme
|7,753,702
|7,409,705
|4.44%
|~500
|
|
| style="text-align:left" |lr_enum_continue 10000000 (10M steps)
|-
| style="text-align:left" |Andrew Ducharme
|7,409,705
|7,192,937
|2.93%
|1858.9
|
|
| style="text-align:left" |Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000 --block-mult=12 --time=1 --tape-limit=500 --lin-steps=0 --no-ctl
|-
| style="text-align:left" |Andrew Ducharme
|7,192,937
|6,711,936
|6.69%
|3.6
|
|
| style="text-align:left" | FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 2
FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 4
|-
| style="text-align:left" |Andrew Ducharme
|6,711,936
|6,506,888
|3.05%
|~500
|
|
| style="text-align:left" |
FAR CPS_LRU maxT 100000 LRUH [1,2] remaining parameters
|-
| style="text-align:left" |Andrew Ducharme
|6,506,888
|6,298,166
|3.21%
|~2200
|
|
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH [3,4]
|-
|Andrew Ducharme
|6,298,166
|6,257,722
|0.64%
|~2000
|
|
|FAR CPS_LRU maxT 100000 LRUH 5
|-
|Andrew Ducharme
|6,257,722
|6,237,675
|0.32%
|~2400
|
|
|FAR CPS_LRU maxT 100000 LRUH 6
|-
|Andrew Ducharme
|6,237,675
|6,156,619
|1.30%
|1798.3
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=250_000 --block-mult=1 --time=1 --tape-limit=1000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|6,156,619
|6,123,679
|0.54%
|1784.3
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=250_000 --block-mult=5 --time=1 --tape-limit=1000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|6,123,679
|6,071,297
|0.86%
|~7500
|
|
|FAR CPS_LRU maxT 100000 LRUH 12
|-
|Andrew Ducharme
|6,071,297
|5,913,070
|2.61%
|~25000
|
|
|FAR CPS_LRU maxT 1000000 LRUH [1,2]
|-
|Andrew Ducharme
|5,913,070
|5,718,346
|3.29%
|15790.2
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=1_000_000 --block-mult=8 --time=10 --tape-limit=5000 --max-steps-per-macro=1_000_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|5,718,346
|5,641,006
|1.35%
|15989.4
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=10_000_000 --block-mult=3 --time=10 --tape-limit=5000 --max-steps-per-macro=1_000_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|5,641,006
|5,127,263
|9.11%
|94.3
|
|
|Inductive decider maxT 1000 --exploop
|-
| style="text-align:center" |'''Cumulative'''
|'''9,401,447'''
|'''5,127,263'''
|'''45.46%'''
| ---
| ---
| ---
| style="text-align:left" | ---
|}

==References==

[[Category:BB Domains]][[Category:BB(4,3)]]

TMBR: April 2026

2026-05-04T17:36:47Z

ADucharme: /* Holdouts */ add new 4x3 top 10 scorers

{{TMBRnav|March 2026|May 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

[[:Category:This Month in Beaver Research|This Month in Beaver Research]] for April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to the [[BMO|Beaver Math Olympiad]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]]. There was a 40% reduction in [[BB(4,3)]], and we also passed 18 million holdouts for [[BB(7)]]. There's been a lot of discoveries in the [[Fractran]], [[GRF|General Recursive Function]], and [[Lambda Calculus]] versions of Busy Beaver. Katelyn Doucette created a visualizer for Fractran space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir family]] of Cryptids.[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>] The first BBµ champion was found that takes advantage of the minimization (M) operator. GRF Cryptids of sizes as small as 56 were found. Both BBf(100) and BBµ(100) were proven to surpass [[Graham's number]]. BBλ(38) was solved and a 74-bit BBλ Cryptid was found.

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir family]]. Enumeration of BBf(23) would take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>]
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a series of fast-growing programs: a tetrational program of size 29,[https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330] <math>f_\omega</math> programs starting from size 86,[https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789] and <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** A number of [[Cryptids]] were hand-constructed: size 141, by Jacob on 8 Apr,[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>] size 81, by Shawn Ligocki on 28 Apr,[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf] and size 56, by Shawn on 2 May.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o Min) up to size 20.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>5\cdot{2^{2^{2^{2^2}}}} + 6</math>).
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits].
**Tromp's BB Lambda paper got published in the journal [https://www.mdpi.com/1099-4300/28/5/494 Entropy].
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Although it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Misc ==

* ZTS439 explored some properties of summations over the [[Hydra function]] <math>S(n) = \sum_{k=0}^n H(k)</math>.[https://discord.com/channels/960643023006490684/1497472476215640174/1497472476215640174]

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[11]</nowiki>] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a number theory result which would be a major project in of itself to formalise. Therefore, the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from finding new equivalences. This means there is now only 1 holdout (see above) whose solution has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**[https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 Alistaire] and Discord user [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 @The_Real_Fourious_Banana] each simulated a TM to 1e15 steps. Combined with the recent equivalence reductions (10 machines total), the number of machines to simulate to 1e14 and 1e15 steps is 161 & 225 respectively.
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235 <nowiki>[12]</nowiki>] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction. He also found several new high-scoring halters, current places 4 through 8 in the 4x3 Busy Beaver game. 4th place is {{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}} with approximate sigma score ~10↑↑1023.47221. [https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143 <nowiki>[13]</nowiki>]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim.[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[15]</nowiki>] These TMs, in addition to {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}}, were shown to halt in 2024 June (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]), but step counts and scores for these machines were unknown.
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[18]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[19]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[20]</nowiki>]

[[Category:This Month in Beaver Research|2026-04]]

User:ADucharme

2026-05-04T17:31:59Z

ADucharme: add BB(3,4)

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts remaining from that initial push. Specifically, my enumeration work has solved

* 67.27% of the last 86,129,304, and 12.65% of the last 20,405,295, BB(7) holdouts
* 44.74% of the last 970,101 BB(2,6) holdouts
* 3.10% of the last 12,435,284 BB(3,4) holdouts
* 40.00% of the last 9,401,447 BB(4,3) holdouts

I've also tried pen-and-paper analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively-studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last ~1500 BB(6) holdouts, I solved 67 and counting. Partial credit for some of these machines goes to Peacemaker II, who identifies permutations of machines I solved in the holdout list. Because of the shared behavior between permutations, I can apply the decider which solved to the original TM I found to the permutations, and often solve permutations too.

Solved halting TMs (49) with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs (18) with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9
1RB0RB_1LC1RA_0LA1RD_1LA1LE_1LF1LD_---0LC FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 3 n 12
1RB0LD_1RC0RE_0LA0RC_1LA1LD_0RF1RA_---1RC FAR CPS_LRU maxT 10000000 LRUH 6 H 3 tH 0 n 9
1RB1LB_1LC1RE_0RD0LB_0LB1RA_1LA0RF_---0RC FAR CPS_LRU maxT 10000000 LRUH 7 H 3 tH 1 n 4
1RB0LD_0RC1RF_1RD0RA_1LE1RB_1LC0LE_1RC--- FAR CPS_LRU maxT 10000000 LRUH 7 H 4 tH 1 n 24
1RB0LA_0RC---_1RD1RE_1LA1LD_1RD0RF_0RC1RC FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB0LA_1RC1RA_0LD1LA_1LF1RE_0RD0RE_0LC--- FAR RWL_mod maxT 10000000 H 4 mod 1 n 8
1RB1RF_1LC1LB_---0LD_1RE0LD_0RA1RA_0LE0RE FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB1RD_0RC1RE_1LD0RE_1LB---_0RA1LF_0LE0LF FAR CPS_LRU maxT 1000000 LRUH 32 H 1 tH 29 n 12
1RB1RE_1LC0RF_1RE0LD_1LC0LB_1RA0RE_1RC--- FAR CPS_LRU maxT 1000000 LRUH 32 H 4 tH 20 n 24
1RB0RE_1LC1RA_0LA1LD_1RE1LC_0RF1RB_---0LC FAR CPS_LRU maxT 1000000 LRUH 17 H 4 tH 13 n 3

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Games ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain. These remain my best-ever entries in these particular Busy Beaver games.

==== BB(7) ====
{| class="wikitable sortable"
|Place
|TM
|Score
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|5
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG}}
|10 ↑↑ 403.84
|-
|9
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}}
|10 ↑↑ 243.88
|}

{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}} was a bit of co-discovery: Iijil first enumerated the TM and I first showed it was halting.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}

==== BB(4,3) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|4
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|~10 ↑↑ 1023.47221
|-
|5
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 619.07737
|-
|6
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|~10 ↑↑ 512.10945
|-
|7
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|~10 ↑↑ 439.02781
|-
|8
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 234.06408
|}

TMBR: April 2026

2026-05-04T02:53:10Z

ADucharme: /* BB Adjacent */ LC copy-editing.

{{TMBRnav|March 2026|May 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

[[:Category:This Month in Beaver Research|This Month in Beaver Research]] for April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to the [[BMO|Beaver Math Olympiad]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]]. There was a 40% reduction in [[BB(4,3)]], and we also passed 18 million holdouts for [[BB(7)]]. There's been a lot of discoveries in the [[Fractran]], [[GRF|General Recursive Function]], and [[Lambda Calculus]] versions of Busy Beaver. Katelyn Doucette created a visualizer for Fractran space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir family]] of Cryptids.[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>] The first BBµ champion was found that takes advantage of the minimization (M) operator. GRF Cryptids of sizes as small as 56 were found. Both BBf(100) and BBµ(100) were proven to surpass [[Graham's number]]. BBλ(38) was solved and a 74-bit BBλ Cryptid was found.

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir family]]. Enumeration of BBf(23) would take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>]
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a series of fast-growing programs: a tetrational program of size 29,[https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330] <math>f_\omega</math> programs starting from size 86,[https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789] and <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** A number of [[Cryptids]] were hand-constructed: size 141, by Jacob on 8 Apr,[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>] size 81, by Shawn Ligocki on 28 Apr,[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf] and size 56, by Shawn on 2 May.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o Min) up to size 20.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>5\cdot{2^{2^{2^{2^2}}}} + 6</math>).
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits].
**Tromp's BB Lambda paper got published in the journal [https://www.mdpi.com/1099-4300/28/5/494 Entropy].
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Although it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Misc ==

* ZTS439 explored some properties of summations over the [[Hydra function]] <math>S(n) = \sum_{k=0}^n H(k)</math>.[https://discord.com/channels/960643023006490684/1497472476215640174/1497472476215640174]

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[11]</nowiki>] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a number theory result which would be a major project in of itself to formalise. Therefore, the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from finding new equivalences. This means there is now only 1 holdout (see above) whose solution has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**[https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 Alistaire] and Discord user [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 @The_Real_Fourious_Banana] each simulated a TM to 1e15 steps. Combined with the recent equivalence reductions (10 machines total), the number of machines to simulate to 1e14 and 1e15 steps is 161 & 225 respectively.
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235 <nowiki>[12]</nowiki>] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143 <nowiki>[13]</nowiki>]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim.[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[15]</nowiki>] These TMs, in addition to {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}}, were shown to halt in 2024 June (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]), but step counts and scores for these machines were unknown.
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[18]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[19]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[20]</nowiki>]

[[Category:This Month in Beaver Research|2026-04]]

TMBR: April 2026

2026-05-04T02:45:21Z

ADucharme: GRF copy editing

{{TMBRnav|March 2026|May 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

[[:Category:This Month in Beaver Research|This Month in Beaver Research]] for April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to the [[BMO|Beaver Math Olympiad]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]]. There was a 40% reduction in [[BB(4,3)]], and we also passed 18 million holdouts for [[BB(7)]]. There's been a lot of discoveries in the [[Fractran]], [[GRF|General Recursive Function]], and [[Lambda Calculus]] versions of Busy Beaver. Katelyn Doucette created a visualizer for Fractran space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir family]] of Cryptids.[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>] The first BBµ champion was found that takes advantage of the minimization (M) operator. GRF Cryptids of sizes as small as 56 were found. Both BBf(100) and BBµ(100) were proven to surpass [[Graham's number]]. BBλ(38) was solved and a 74-bit BBλ Cryptid was found.

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir family]]. Enumeration of BBf(23) would take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>]
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a series of fast-growing programs: a tetrational program of size 29,[https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330] <math>f_\omega</math> programs starting from size 86,[https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789] and <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** A number of [[Cryptids]] were hand-constructed: size 141, by Jacob on 8 Apr,[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>] size 81, by Shawn Ligocki on 28 Apr,[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf] and size 56, by Shawn on 2 May.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o Min) up to size 20.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>5\cdot{2^{2^{2^{2^2}}}} + 6</math>)
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits.]
**Tromp's BB Lambda paper got published: [https://www.mdpi.com/1099-4300/28/5/494 MDPI] -- [https://doi.org/10.3390/e28050494 DOI]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Although it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Misc ==

* ZTS439 explored some properties of summations over the [[Hydra function]] <math>S(n) = \sum_{k=0}^n H(k)</math>.[https://discord.com/channels/960643023006490684/1497472476215640174/1497472476215640174]

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[11]</nowiki>] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a number theory result which would be a major project in of itself to formalise. Therefore, the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from finding new equivalences. This means there is now only 1 holdout (see above) whose solution has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**[https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 Alistaire] and Discord user [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 @The_Real_Fourious_Banana] each simulated a TM to 1e15 steps. Combined with the recent equivalence reductions (10 machines total), the number of machines to simulate to 1e14 and 1e15 steps is 161 & 225 respectively.
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235 <nowiki>[12]</nowiki>] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143 <nowiki>[13]</nowiki>]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim.[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[15]</nowiki>] These TMs, in addition to {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}}, were shown to halt in 2024 June (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]), but step counts and scores for these machines were unknown.
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[18]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[19]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[20]</nowiki>]

[[Category:This Month in Beaver Research|2026-04]]

TMBR: April 2026

2026-05-03T21:37:11Z

ADucharme: copy editing

{{TMBRnav|March 2026|May 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

[[:Category:This Month in Beaver Research|This Month in Beaver Research]] for April 2026. This month, a new [[Cryptid]] was discovered in [[BB(6)]] by Discord user sheep, and [[Beaver Math Olympiad#8. 1RB0LD 0RC1RB 0RD0RA 1LE0RD 1LF--- 0LA1LA (bbch)|BMO 8]] was added to the [[BMO|Beaver Math Olympiad]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]]. There was a 40% reduction in [[BB(4,3)]], and we also passed 18 million holdouts for [[BB(7)]]. There's been a lot of discoveries in [[Fractran]], General Recursive Functions ([[GRF]]), and [[Lambda Calculus]] versions of Busy Beaver. Katelyn Doucette created a visualizer for Fractran space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir family]] of Cryptids.[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>] The first BBµ champion was found that takes advantage of the minimization (M) operator. GRF Cryptids of sizes as small as 56 were found. Both BBf(100) and BBµ(100) were proven to surpass [[Graham's number]]. BBλ(38) was solved and a 74-bit BBλ Cryptid was found.

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.|500x500px]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir family]]. Enumeration of BBf(23) would take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>]
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a series of fast-growing programs: a tetrational program of size 29,[https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330] <math>f_\omega</math> programs starting from size 86,[https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789] and <math>f_{\omega + 1}</math> programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
* [[General Recursive Function]]
** Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** A number of [[Cryptids]] were hand-constructed: size 141, by Jacob on 8 Apr,[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>] size 81, by Shawn Ligocki on 28 Apr,[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/erdos.mgrf] and size 56, by Shawn on 2 May.[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Shawn built an "[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/ack_worm.mgrf Ackermann worm]" function with <math>f_{\omega}</math> growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** Jacob extended the Ackermann worm to find a <math>f_{\omega^2}</math> growth function of size 204 on 23 Apr.[https://discord.com/channels/960643023006490684/1447627603698647303/1497037415628411082][https://discord.com/channels/960643023006490684/1447627603698647303/1497257739850879106]
** Shawn enumerated all Primitive Recursive Functions (GRF w/o Min) up to size 20.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1497797672742944898]
** Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[https://discord.com/channels/960643023006490684/1447627603698647303/1499746900860211214]
** Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[https://discord.com/channels/960643023006490684/1447627603698647303/1498743904433082379]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BBλ(38) has been solved] (BBλ(38) = <math>5\cdot{2^{2^{2^{2^2}}}} + 6</math>)
**[https://discord.com/channels/960643023006490684/1355653587824283678/1493455967868817429 A Cryptid was found in 74 bits.]
**Tromp's BB Lambda paper got published: [https://www.mdpi.com/1099-4300/28/5/494 MDPI] -- [https://doi.org/10.3390/e28050494 DOI]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Although it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Misc ==

* ZTS439 explored some properties of summations over the [[Hydra function]] <math>S(n) = \sum_{k=0}^n H(k)</math>.[https://discord.com/channels/960643023006490684/1497472476215640174/1497472476215640174]

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1161
|1104
|57
|4.91%
|-
|[[BB(7)]]
|18,036,852
|17,823,260
|213,592
|1.18%
|-
|[[BB(4,3)]]
|9,401,447
|5,641,006
|3,760,441
|40.00%
|-
|[[BB(3,4)]]
|12,435,284
|12,049,358
|385,926
|3.10%
|-
|[[BB(2,5)]]
|69
|66
|3
|4.35%
|-
|[[BB(2,6)]]
|545,005
|536,112
|11,241
|1.63%
|}
[[File:BB6 progress Q1 2026.png|alt=BB(6) progress in 2026 so far -- by mxdys|thumb|521x521px|BB(6) progress in 2026 so far -- by mxdys]]
*[[BB(6)]]: Reduction: '''57'''. No. of TMs to simulate to 1e14: '''161''' (reduction: 10). To 1e15: '''225''' (reduction: 13).
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[11]</nowiki>] a new [[Cryptid]], {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, similar to [[Space Needle]]. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
**BMO 8 was added to the [[Beaver Math Olympiad]]: {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA}}
**The Turing Machine <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code> has been informally solved for months now. The formal solution depends on a number theory result which would be a major project in of itself to formalise. Therefore, the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See [https://github.com/rwst/bbchallenge/blob/main/1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---/Bootstrap.lean Github], Axiom minimal version: [https://discord.com/channels/960643023006490684/1443295684878143579/1494887513888657605 Discord], The machine's Discord thread: [https://discord.com/channels/960643023006490684/1443295684878143579/1495013820098150450 Link]. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1497651809773289552 released] a new holdouts list of '''1119''' machines, the reduction mostly (except for [https://discord.com/channels/960643023006490684/1239205785913790465/1497668636117176520 one TM], the other informal holdout) came from finding new equivalences. This means there is now only 1 holdout (see above) whose solution has not been fully formalised.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**These equivalences were found with the help of -d, see (Discord [https://discord.com/channels/960643023006490684/960643023530762341/1498924022182973561 1], [https://discord.com/channels/960643023006490684/960643023530762341/1498732973086998739 2], [https://discord.com/channels/960643023006490684/1239205785913790465/1499331999599558656 3]). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
**[https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 Alistaire] and Discord user [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 @The_Real_Fourious_Banana] each simulated a TM to 1e15 steps. Combined with the recent equivalence reductions (10 machines total), the number of machines to simulate to 1e14 and 1e15 steps is 161 & 225 respectively.
*[[BB(7)]]
**Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to '''17,823,260'''.[https://discord.com/channels/960643023006490684/1369339127652159509/1490808711952728235 <nowiki>[12]</nowiki>] (A '''1.18%''' reduction)
* [[BB(4,3)]]:
** In [[BB(4,3)#Stage 3|phase 2 stage 3]], Andrew Ducharme reduced the number of holdouts from 9,401,447 to '''5,641,006''', a '''40.00%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1497715882049147143 <nowiki>[13]</nowiki>]
* [[BB(3,4)]]:
**Andrew Ducharme began [[BB(3,4)#Phase 3|Phase 3]], reducing the holdout count from 12,435,284 to '''12,049,358''' (a '''3.10%''' reduction) with mxdys's FAR decider.
* [[BB(2,5)]]:
** On 1 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution], which he created using [https://aristotle.harmonic.fun/ Aristotle AI]. Then [https://discord.com/channels/960643023006490684/1259770421046411285/1488898494386274374 mxdys formalised the result] in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
** On 2 April 2026, [https://discord.com/channels/960643023006490684/1259770421046411285/1489095097373954199 mxdys solved] [[Beaver Math Olympiad#Solved problems|BMO 3]] variant {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB}} using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
** {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|halt}} and {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}} were simulated until halting by prurq using Quick_Sim.[https://discord.com/channels/960643023006490684/1259770421046411285/1492999358482874448 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[15]</nowiki>] These TMs, in addition to {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}}, were shown to halt in 2024 June (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]), but step counts and scores for these machines were unknown.
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''536,112''' via Enumerate.py, a '''1.63%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[18]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to '''687,123,946'''. A total of 220K subtasks out of the 1 million subtasks (or '''22%''') have been enumerated. (see [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]) [https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[19]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[20]</nowiki>]

[[Category:This Month in Beaver Research|2026-04]]

BB(2,6)

2026-05-02T19:34:29Z

ADucharme: /* Top Halters */ new 9th place halter

The 2-state, 6-symbol Busy Beaver problem, '''BB(2,6),''' is unsolved. With cryptids like [[Hydra]] in the preceding domain [[BB(2,5)]], we know that we must solve a [[Collatz-like]] problem in order to solve BB(2,6).

The current BB(2,6) champion {{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}} was discovered by Pavel Kropitz in May 2023, proving the lower bound:<math display="block">S(2,6) > \Sigma(2,6) > 10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}} > 10 \uparrow \uparrow \uparrow 3</math>

== 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 20 highest known scoring machines are:
{| class="wikitable"
|+
!TM
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|10 ↑↑↑ 3
|Pavel Kropitz
|-
|{{TM|1RB2LA1RZ1RB5RB0RB_2LA4RA3LB5LB5RA4LB|halt}}
|10 ↑↑ 19892.08
|Peacemaker II
|-
|{{TM|1RB3LA4LB0RB1RA3LA_2LA2RA4LA1RA5RB1RZ|halt}}
|10 ↑↑ 91.17
|Pavel Kropitz
|-
|{{TM|1RB2LA1RA4LA5RA0LB_1LA3RA2RB1RZ3RB4LA|halt}}
|10 ↑↑ 70.27
|Shawn Ligocki
|-
|{{TM|1RB2LB1RZ3LA2LA4RB_1LA3RB4RB1LB5LB0RA|halt}}
|10 ↑↑ 69.68
|Shawn Ligocki
|-
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|Andrew Ducharme
|-
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|Andrew Ducharme
|-
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|Andrew Ducharme
|-
|{{TM|1RB2LA5LB0RA1RA3LB_1LA4LA3LB3RB3RB1RZ|halt}}
|10 ↑↑ 23.9964
|Andrew Ducharme
|-
|{{TM|1RB3LB3RB4LA2LA4LA_2LA2RB1LB0RA5RA1RZ|halt}}
|10 ↑↑ 21.54
|Shawn Ligocki
|-
|{{TM|1RB2LB3LA1RA0RA1RZ_1LA2RB1LB4RB5RA3LA|halt}}
|10 ↑↑ 20.58
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB1RA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB5LA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Andrew Ducharme
|-
|{{TM|1RB3RA4LB5RA5LB4RA_2LA1RZ1RB2LA5LA0LA|halt}}
|10 ↑↑ 17.08
|Andrew Ducharme
|-
|{{TM|1RB3RA4LA1LA0LA1RZ_2LA0LB1RA1LB5LB2RA|halt}}
|10 ↑↑ 15.44
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ2LA|halt}}
|10 ↑↑ 14.35
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ3RA|halt}}
|10 ↑↑ 14.17
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ1LA|halt}}
|10 ↑↑ 14.05
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ0RA|halt}}
|10 ↑↑ 13.69
|Andrew Ducharme
|-
|{{TM|1RB3LA3RA4LB2LB0LA_2LA5LB2RB0RA0RA1RZ|halt}}
|10 ↑↑ 12.42
|Andrew Ducharme
|}
All decimal places are truncated.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in November 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~24B(illion) TMs of which ~2.278B were holdout TMs. This was reduced to ~22M holdout TMs (a 99.02% reduction). The details are given in this table, including links to the Google Drive with the holdouts and details of the computation:

(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"
!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" |Data
|-
|style="text-align:left" |Terry Ligocki
|2,278,655,696
|2,109,114,609
|7.44%
|40.9
|1,150.90
|15,468.23
|style="text-align:left" |Reverse_Engineer_Filter.py
|style="text-align:left", rowspan="100" |[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|2,109,114,609
|683,067,538
|67.61%
|452.8
|874.77
|1,293.79
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|683,067,538
|210,993,434
|69.11%
|396.4
|330.85
|478.72
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|210,993,434
|141,680,232
|32.85%
|273.9
|70.29
|213.97
|style="text-align:left" |CPS_Filter.py --block-size=3 --max_steps=10_000
|-
|style="text-align:left" |Terry Ligocki
|141,680,232
|66,029,536
|53.40%
|486.6
|43.18
|80.87
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --time=10 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|66,029,536
|46,119,004
|30.15%
|167.4
|33.05
|109.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.01 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|46,119,004
|39,034,142
|15.36%
|170.1
|11.57
|75.34
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=12 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|39,034,142
|29,109,512
|25.43%
|2,221.6
|1.24
|4.88
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:left" |Terry Ligocki
|29,109,512
|24,536,819
|15.71%
|384.2
|3.31
|21.05
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=6 --recursive --no-steps --time=0.05 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|24,536,819
|22,302,296
|9.11%
|1,047.5
|0.59
|6.51
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=4 --recursive --no-steps --time=1.00 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|}

== Phase 2 ==
When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===
Andrew Ducharme ran another pass of "lr_enum_continue" with the maximum number of steps set to 10 million. The holdouts were reduced from ~22.3M TMs to ~20.4M TMs (a 8.72% reduction). The entry in the table below has a rather technical/arcane/cryptic description. This was an effort to capture enough information to rerun that filter in parallel with specific C++ code, lr_enum_continue, and a specific parallel queuing system, Slurm:

(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"
!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" |Data
|-
|style="text-align:left" |Andrew Ducharme
|22,302,296
|20,358,011
|8.72%
|1,350.0
|0.40
|4.59
|style="text-align:left" |lr_enum_continue ${WORK_DIR}chunk_${SLURM_ARRAY_TASK_ID} 10000000 ${WORK_DIR}halt_${SLURM_ARRAY_TASK_ID}.txt ${WORK_DIR}inf_${SLURM_ARRAY_TASK_ID}.txt ${WORK_DIR}unknown_${SLURM_ARRAY_TASK_ID}.txt "" false
|style="text-align:left" rowspan="50"|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|}

=== Stage 2 ===
Starting from the results of Stage 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 50 variations were run. The holdouts were reduced from ~20.4M TMs to ~907K TMs (a 95.5% reduction). The details are given in this table, including links to the Google Drive with the holdouts and details of the computation:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|20,358,011
|19,500,847
|4.21%
|22.0
|10.84
|257.42
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 6 run
|style="text-align:left" rowspan="50"|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|19,500,847
|18,747,861
|3.86%
|86.0
|2.43
|63.01
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|18,747,861
|4,811,076
|74.34%
|47.0
|82.33
|110.75
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|4,811,076
|2,982,075
|38.02%
|17.1
|29.74
|78.22
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|2,982,075
|2,897,340
|2.84%
|15.2
|1.55
|54.64
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 8 mod 3 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|2,897,340
|2,850,781
|1.61%
|16.7
|0.77
|48.17
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n 7 run
|-
|style="text-align:left" |Terry Ligocki
|2,850,781
|2,759,635
|3.20%
|13.7
|1.85
|58.01
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|2,759,635
|1,953,426
|29.21%
|13.6
|16.48
|56.42
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,953,426
|1,855,545
|5.01%
|2.4
|11.18
|223.14
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,855,545
|1,647,269
|11.22%
|6.6
|8.80
|78.40
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,647,269
|1,608,166
|2.37%
|3.4
|3.20
|134.96
|style="text-align:left" |chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,608,166
|1,585,745
|1.39%
|9.6
|0.65
|46.35
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|1,585,745
|1,555,673
|1.90%
|5.7
|1.47
|77.73
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 10000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,555,673
|1,428,534
|8.17%
|9.3
|3.78
|46.31
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 2 tH 0 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,428,534
|1,340,964
|6.13%
|0.8
|29.70
|484.55
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 10000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,340,964
|1,286,439
|4.07%
|0.8
|18.40
|452.56
|style="text-align:left" |chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,286,439
|1,273,911
|0.97%
|0.8
|4.20
|430.88
|style="text-align:left" |chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,273,911
|1,265,198
|0.68%
|0.8
|2.88
|420.73
|style="text-align:left" |chr_LRUH 3 chr_H 1 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,265,198
|1,258,925
|0.50%
|0.9
|1.99
|400.83
|style="text-align:left" |chr_LRUH 8 chr_H 6 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,258,925
|1,242,136
|1.33%
|0.8
|5.51
|412.84
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 0 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,242,136
|1,231,731
|0.84%
|1.0
|2.78
|331.77
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 2 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,231,731
|1,216,646
|1.22%
|1.0
|4.15
|338.72
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 12 H 0 tH 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,216,646
|1,214,294
|0.19%
|0.9
|0.76
|393.03
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 2 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,214,294
|1,213,431
|0.07%
|0.9
|0.28
|391.30
|style="text-align:left" |chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,213,431
|1,211,390
|0.17%
|1.1
|0.52
|307.13
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 1 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,211,390
|1,209,989
|0.12%
|1.1
|0.35
|306.09
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,209,989
|1,209,974
|0.00%
|0.9
|0.00
|381.42
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 16 H 1 tH 0 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,209,974
|1,201,890
|0.67%
|2.5
|0.90
|134.19
|style="text-align:left" |chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,201,890
|1,200,086
|0.15%
|1.3
|0.37
|248.36
|style="text-align:left" |chr_LRUH 10 chr_H 6 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,200,086
|1,199,734
|0.03%
|1.2
|0.08
|270.32
|style="text-align:left" |chr_asth 0 chr_LRUH 3 chr_H 3 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,199,734
|1,198,893
|0.07%
|2.3
|0.10
|147.66
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 2 mod 6 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,198,893
|1,165,493
|2.79%
|4.5
|2.05
|73.44
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 4 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,165,493
|1,153,863
|1.00%
|9.3
|0.35
|34.88
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 0 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,153,863
|1,144,711
|0.79%
|3.7
|0.69
|87.51
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 5 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,144,711
|1,127,789
|1.48%
|7.9
|0.60
|40.26
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 30000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,127,789
|1,124,762
|0.27%
|4.7
|0.18
|66.75
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 0 tH 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|1,124,762
|1,117,226
|0.67%
|5.6
|0.37
|55.36
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 12 H 0 tH 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,117,226
|1,109,057
|0.73%
|7.7
|0.30
|40.49
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,109,057
|1,083,097
|2.34%
|11.4
|0.63
|27.06
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 30000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,083,097
|1,077,833
|0.49%
|11.2
|0.13
|26.81
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 100000 NG_n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,077,833
|1,066,795
|1.02%
|24.1
|0.13
|12.40
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 2 tH 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,066,795
|1,039,229
|2.58%
|52.6
|0.15
|5.64
|style="text-align:left" |chr_LRUH 14 chr_H 6 MitM_CTL NG maxT 100000 NG_n 11 run
|-
|style="text-align:left" |Terry Ligocki
|1,039,229
|1,019,286
|1.92%
|43.5
|0.13
|6.63
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 12 mod 1 n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,019,286
|993,556
|2.52%
|66.8
|0.11
|4.24
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 8 H 2 tH 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|993,556
|985,718
|0.79%
|78.3
|0.03
|3.53
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 1 tH 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|985,718
|981,095
|0.47%
|83.7
|0.02
|3.27
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 1 tH 0 n 9 run
|-
|style="text-align:left" |Terry Ligocki
|981,095
|975,912
|0.53%
|79.4
|0.02
|3.43
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 16 mod 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|975,912
|974,180
|0.18%
|84.6
|0.01
|3.20
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 16 mod 4 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|974,180
|971,254
|0.30%
|96.9
|0.01
|2.79
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 12 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|971,254
|970,101
|0.12%
|105.6
|0.00
|2.56
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 0 tH 0 n 18 run
|}

=== Stage 3 ===
Starting from the results of Stage 2, Andrew Ducharme ran "lr_enum_continue" with the maximum number of steps set to 100 million, then "Enumerate.py" with various parameters. A total of 10 Enumerate variations were run. The holdouts were reduced from ~970K TMs to ~867K TMs (a 10.63% reduction). The details are given in this table, including links to the Google Drive with the holdouts and details of the computation:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Andrew Ducharme
|970,101
|939,447
|3.16%
| --
| --
| --
|style="text-align:left" |lr_enum_continue 100_000_000 steps
| rowspan="11" |[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|-
| style="text-align:left" |Andrew Ducharme
|939,447
|903,224
|3.86%
|440.3
|0.03
|0.59
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=4 --no-ctl --lin-steps=0 --time=2 --force --save-freq=1000
|-
| style="text-align:left" |Andrew Ducharme
|903,224
|895,813
|0.82%
|647.7
|0.00
|0.39
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=3 --no-ctl --lin-steps=0 --time=3 --force --save-freq=1000
|-
| style="text-align:left" |Andrew Ducharme
|895,813
|889,838
|0.67%
|609.3
|0.00
|0.41
| style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=8 --no-ctl --lin-steps=0 --time=4 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|889,838
|880,278
|1.07%
|1,638.9
|0.00
|0.15
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=12 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|880,278
|877,485
|0.32%
|1,885.5
|0.00
|0.13
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=6 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|877,485
|875,062
|0.28%
|2,068.8
|0.00
|0.12
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=5 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|875,062
|873,469
|0.18%
|1,785.4
|0.00
|0.14
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=7 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|873,469
|870,085
|0.39%
|9,270.0
|0.00
|0.03
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=2 --tape-limit=500 --time=120 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|870,085
|869,001
|0.12%
|4,498.3
|0.00
|0.05
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=10_000_000 --block-mult=60 --tape-limit=5000 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|869,001
|867,008
|0.23%
|3997.4
|0.00
|0.06
|style="text-align:left"|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=9 --tape-limit=5000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl --force --save-freq=250
|}
The total time spent on the lr_enum_continue computation was not recorded.

=== Stage 4 ===
Following the release of @mxdys's implementation of FAR deciders in C++, these deciders were applied to the 2x6 holdouts by Andrew Ducharme. The details are given in this table, including links to the Google Drive with the holdouts and solved TMs per decider:

(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"
! colspan="2" |Holdout TMs
! rowspan="2" |<math>*^1</math>
! rowspan="2" |<math>*^2</math>
! colspan="2" |TMs/sec/core
! rowspan="2" |Description
! rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|867,008
|811,301
|6.43%
|0.043
|364.10
|5,666.72
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 2 H 1 tH 1 n 2
| rowspan="28" |[https://drive.google.com/drive/folders/18njhmOzRc67zCmVuLd0aDxl6ETBhL1gy?usp=sharing Google Drive]
|-
|811,301
|806,119
|0.64%
|0.159
|9.03
|1,413.42
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 3 H 1 tH 1 n 2
|-
|806,119
|736,690
|8.61%
|0.548
|35.21
|408.78
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 1 tH 1 n 2
|-
|736,690
|736,504
|0.03%
|0.009
|5.81
|23,021.56
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 1 H 1 tH 1 n 1
|-
|736,504
|735,317
|0.16%
|0.058
|5.71
|3,540.88
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 2
|-
|735,317
|733,717
|0.22%
|0.341
|1.30
|599.28
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 2 tH 2 n 2
|-
|733,717
|673,920
|8.15%
|3.8
|4.43
|54.32
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 2 tH 2 n 4
|-
|673,920
|652,828
|3.13%
|~10
| ---
| ---
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 6 H 2 tH 2 n 4
|-
|652,828
|645,264
|1.16%
|~12
| ---
| ---
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 8 H 2 tH 2 n 4
|-
|645,264
|641,388
|0.60%
|~15
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 100000 LRUH 10 H 2 tH 2 n 10
|-
|641,388
|635,505
|0.92%
|~200
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 10 H 1 tH 2 n 10
|-
|635,505
|616,639
|2.97%
| ---
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 2 H 0 tH 0 n [3-10]
|-
|616,639
|592,039
|3.99%
|~700
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 3 H 0 tH 0 n [1-10]
|-
|592,039
|576,938
|2.55%
|~800
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 3 H [0-1] tH [0-1] n [1-10]
|-
|576,938
|572,963
|0.69%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4 H 0 tH 0 n [1-10]
|-
|572,963
|567,971
|0.87%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4 H 2 tH 0 n [1-10]
|-
|567,971
|566,096
|0.33%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 6 H 0 tH 0 n [1-10]
|-
|566,096
|564,290
|0.32%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 0 tH [0,2] n [1-10]
|-
|564,290
|559,553
|0.84%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 2 tH 1 n [1-10]
|-
|559,553
|558,039
|0.27%
|~900
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 2 tH 2 n [1-10]
|-
|558,039
|556,814
|0.22%
|~14,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH [12,16] H [0-2] tH [0-2] n [1-10]
|-
|556,814
|554,479
|0.42%
|~3,600
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH [1-3]
|-
|554,479
|551,586
|0.52%
|~5000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4
|-
|551,586
|548,993
|0.47%
|~13,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 5
|-
|548,993
|545,005
|0.73%
|~57,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 6 and 8
|-
|545,005
|542,325
|0.49%
|6851.2
|0.00
|0.022
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=96 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|-
|542,325
|537,393
|0.91%
|9032.1
|0.00
|0.017
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=2 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|-
|537,393
|536,112
|0.24%
|8969.4
|0.00
|0.017
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=3 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|}

== References ==


[[Category: BB Domains]][[Category:BB(2,6)]]

Beaver Math Olympiad

2026-04-28T17:14:10Z

ADucharme: /* 6. 1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD (bbch) */ rename section to Space Needle like how BMO 2 is "Hydra and Antihydra"

'''Beaver Mathematical Olympiad''' (BMO) is an attempt to re-formulate the halting problem for some particular Turing machines as a mathematical problem in a style suitable for a hypothetical math olympiad.

The purpose of the BMO is twofold. First, statements where non-essential details (related to tape encoding, number of steps, etc.) are discarded are more suitable to be shared with mathematicians who perhaps are able to help. Second, it's a way to jokingly highlight how a hard question could appear deceptively simple.

BMO problems have been formalized in Lean and added to the DeepMind formal-conjectures database ([https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/Other/BeaverMathOlympiad.lean Github]).

== Unsolved problems ==

=== 1. {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}} ===

Let <math>(a_n)_{n \ge 1}</math> and <math>(b_n)_{n \ge 1}</math> be two sequences such that <math>(a_1, b_1) = (1, 2)</math> and

<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}
(a_n-b_n, 4b_n+2) & \text{if }a_n \ge b_n \\
(2a_n+1, b_n-a_n) & \text{if }a_n < b_n
\end{cases}</math>

for all positive integers <math>n</math>. Does there exist a positive integer <math>i</math> such that <math>a_i = b_i</math>?

The first 10 values of <math>(a_n, b_n)</math> are <math>(1, 2), (3, 1), (2, 6), (5, 4), (1, 18), (3, 17), (7, 14), (15, 7), (8, 30), (17, 22)</math>.

=== 2. [[Hydra]] and [[Antihydra]] ===

Let <math>(a_n)_{n \ge 0}</math> be a sequence such that <math>a_{n+1} = a_n+\left\lfloor\frac{a_n}{2}\right\rfloor</math> for all non-negative integers <math>n</math>.

# If <math>a_0=3</math>, does there exist a non-negative integer <math>k</math> such that the list of numbers <math>a_0, a_1, a_2, \dots, a_k</math> have more than twice as many even numbers as odd numbers? ([[Hydra]])
# If <math>a_0=8</math>, does there exist a non-negative integer <math>k</math> such that the list of numbers <math>a_0, a_1, a_2, \dots, a_k</math> have more than twice as many odd numbers as even numbers? ([[Antihydra]])

=== 5. {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}} ===

Let <math>(a_n)_{n \ge 1}</math> and <math>(b_n)_{n \ge 1}</math> be two sequences such that <math>(a_1, b_1) = (0, 5)</math> and

<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}
(a_n+1, b_n-f(a_n)) & \text{if } b_n \ge f(a_n) \\
(a_n, 3b_n+a_n+5) & \text{if } b_n < f(a_n)
\end{cases}</math>

where <math>f(x)=10\cdot 2^x-1</math> for all non-negative integers <math>x</math>.

Does there exist a positive integer <math>i</math> such that <math>b_i = f(a_i)-1</math>?

=== 6. [[Space Needle]] ===
Let <math>f(b) = b + k + 3a</math>, where <math>k</math> and <math>a</math> are non-negative integers satisfying <math>b = (2a+1)\cdot 2^k</math>.

Now consider the iterated application of the function <math>f^{n+1}(b) = f(f^n(b))</math>, <math>f^0(b)=b</math>. Does there exist a non-negative integer <math>n</math> such that <math>f^n(6)</math> equals a power of 2?

=== 8. {{TM|1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA|undecided}} ===

Let <math>(a_n)_{n \ge 1}</math> and <math>(b_n)_{n \ge 1}</math> be two sequences such that <math>(a_1, b_1) = (10, 12)</math> and

<math display="block">(a_{n+1}, b_{n+1}) = \begin{cases}
(a_n-\left\lfloor\frac{b_n}{2}\right\rfloor-3, 3\left\lfloor\frac{b_n+1}{2}\right\rfloor+6) & \text{if }a_n > \left\lfloor\frac{b_n}{2}\right\rfloor \\
(3a_n+5, b_n-2a_n) & \text{if }a_n \le \left\lfloor\frac{b_n}{2}\right\rfloor
\end{cases}</math>

for all positive integers <math>n</math>. Does there exist a positive integer <math>i</math> such that <math>a_n = \left\lfloor \frac{b_n}{2} \right\rfloor + 1</math>?

== Solved problems ==

=== 3. {{TM|1RB0RB3LA4LA2RA_2LB3RA---3RA4RB|non-halt}} and {{TM|1RB1RB3LA4LA2RA_2LB3RA---3RA4RB|non-halt}} ===

Let <math>v_2(n)</math> be the largest integer <math>k</math> such that <math>2^k</math> divides <math>n</math>. Let <math>(a_n)_{n \ge 0}</math> be a sequence such that

<math display="block">a_n = \begin{cases}
2 & \text{if } n=0 \\
a_{n-1}+2^{v_2(a_{n-1})+2}-1 & \text{if } n \ge 1
\end{cases}</math>

for all non-negative integers <math>n</math>. Is there an integer <math>n</math> such that <math>a_n=4^k</math> for some positive integer <math>k</math>?

Link to Discord discussion: https://discord.com/channels/960643023006490684/1084047886494470185/1252634913220591728

Formalised solution: [https://discord.com/channels/960643023006490684/1259770421046411285/1488737894943166604 Initial announcement], [https://discord.com/channels/960643023006490684/1259770421046411285/1488743526882738276 Lean proof], [https://discord.com/channels/960643023006490684/1259770421046411285/1488781537699696821 LLM-translated Rocq proof], [https://discord.com/channels/960643023006490684/1259770421046411285/1488898995865784442 Proof of closure of existing mid-level rules].

=== 4. {{TM|1RB3RB---1LB0LA_2LA4RA3LA4RB1LB|non-halt}} ===

Bonnie the beaver was bored, so she tried to construct a sequence of integers <math>\{a_n\}_{n \ge 0}</math>. She first defined <math>a_0=2</math>, then defined <math>a_{n+1}</math> depending on <math>a_n</math> and <math>n</math> using the following rules:

* If <math>a_n \equiv 0\text{ (mod 3)}</math>, then <math>a_{n+1}=\frac{a_n}{3}+2^n+1</math>.
* If <math>a_n \equiv 2\text{ (mod 3)}</math>, then <math>a_{n+1}=\frac{a_n-2}{3}+2^n-1</math>.

With these two rules alone, Bonnie calculates the first few terms in the sequence: <math>2, 0, 3, 6, 11, 18, 39, 78, 155, 306, \dots</math>. At this point, Bonnie plans to continue writing terms until a term becomes <math>1\text{ (mod 3)}</math>. If Bonnie sticks to her plan, will she ever finish?

<div class="toccolours mw-collapsible mw-collapsed">'''Solution'''<div class="mw-collapsible-content">
How to guess the closed-form solution: Firstly, notice that <math>a_n \approx \frac{3}{5} \times 2^n</math>. Secondly, calculate the error term <math>a_n - \frac{3}{5} \times 2^n</math>. The error term appears to have a period of 4. This leads to the following guess:

<math display="block">a_n=\frac{3}{5}\begin{cases}
2^n+\frac{7}{3} &\text{if } n\equiv 0 \pmod{4}\\
2^n-2 &\text{if } n\equiv 1 \pmod{4}\\
2^n+1 &\text{if } n\equiv 2 \pmod{4}\\
2^n+2 &\text{if } n\equiv 3 \pmod{4}
\end{cases}</math>

This closed-form solution can be proven correct by induction. Unfortunately, the induction may require a lot of tedious calculations.

For all <math>k</math>, we have <math>a_{4k} \equiv 2\text{ (mod 3)}</math> and <math>a_{4k+1} \equiv a_{4k+2} \equiv a_{4k+3} \equiv 0\text{ (mod 3)}</math>. Therefore, Bonnie will never finish.
</div></div>

=== 7. {{TM|1RB1RF_1RC0RA_1LD1RC_1LE0LE_0RA0LD_0RB---|non-halt}} ===
Let <math>v_2(n)</math> be the largest integer <math>k</math> such that <math>2^k</math> divides <math>n</math>.

Let <math>f(n) = n+1+(v_2(n+1) \bmod 2)</math>.

Now consider the iterated application of the function <math>f^{n+1}(b) = f(f^n(b)))</math>, <math>f^0(b)=b</math>.

Let <math>(a_n)_{n \ge 0}</math> be a sequence such that <math>a_0=1</math> and <math>a_{n+1} = f^{n+2}\left(\left\lfloor\frac{a_n}{2}\right\rfloor\right)</math> for all non-negative integers <math>n</math>.

Does there exist a non-negative integer <math>k</math> such that <math>a_k</math> is even?

(for simplicity, this question is slightly stronger than the halting problem of this TM)

Link to Discord discussion: https://discord.com/channels/960643023006490684/1421782442213376000/1431483206208852001

== Practice Problems ==
Problems that are not BMO-level, but provide counter-examples to certain [[probvious]] intuition:

* {{TM|1RB0LE_1LC1RA_---1LD_0RB1LF_1RD1LA_0LA0RD}}
* {{TM|1RB0RD_0LC1RA_0RA1LB_1RE1LB_1LF1LB_---1LE}}

[[Category:Individual machines]]

BB(4,3)

2026-04-26T07:02:55Z

ADucharme: /* Stage 3 */ new google drive link

The Busy Beaver problem for 4 states and 3 symbols is unsolved. The existence of [[Cryptids]] in the domain is given by the discovery of [[Bigfoot]] in [[BB(3,3)]]. The current [[Champions#3-Symbol TMs|champion]] is {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} which was discovered by Pavel Kropitz in May 2024 along with 6 other long running machines. It was [[User:Polygon/Page for analyses#1RB1RD1LC 2LB1RB1LC 1RZ1LA1LD 0RB2RA2RD (bbch)|analyzed by Polygon]] in Oct 2025, demonstrating the lower bounds:

<math display="block">S(4,3) > \Sigma(4,3) > 10 \uparrow^{4} 4</math>

== Top Halters ==
The longest running halting BB(4,3) TMs are split amongst two classes: the pentational and hexational TMs found by Pavel Kropitz outlined in the Potential Champions section, and the tetrational TMs found by comprehensive holdout filtering by Terry Ligocki. 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 longest running halters found by Pavel Kropitz are:
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|}
The top 20 scoring halting machines found by comprehensive search are:
{| class="wikitable"
|+
!Standard format
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|~10 ↑↑ 1023.47221
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 619.07737
|Andrew Ducharme
|-
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|~10 ↑↑ 512.10945
|Andrew Ducharme
|-
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|~10 ↑↑ 439.02781
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 234.06408
|Andrew Ducharme
|-
|{{TM|1RB0LC1RC_1LA2RB1LB_1RC2LA0RD_2LB1RZ2LC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LA1RA_1LA0RC1LC_1LC2RB0LD_2RA1RZ2RC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_2RC1LB2RD_1LD2LA0LB_0LA1RZ0LC|halt}}
|~10 ↑↑ 178.48320
|Andrew Ducharme
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA1LB_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB1LD0RC_2LC0RB1RA_1RA0LB1RD_0LA2LA1RZ|halt}}
|~10 ↑↑ 158.81916
|Andrew Ducharme
|-
|{{TM|1RB1RC1RB_1LC0RA2LD_2RA0LD1RZ_0LB2LD1RD|halt}}
|~10 ↑↑ 154.52968
|Andrew Ducharme
|-
|{{TM|1RB1LA1RD_2LA0LC2LD_1RZ2RA2LB_0LC2RC1RA|halt}}
|~10 ↑↑ 147.26175
|Andrew Ducharme
|-
|{{TM|1RB0RB1LC_2LC0LD1RA_2RB2LD1RZ_2LA2LB0LD|halt}}
|~10 ↑↑ 141.44248
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ2LD1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC1LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA---|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|}

== Potential Champions ==
In May 2024, [https://discord.com/channels/960643023006490684/1026577255754903572/1243253180297646120 Pavel Kropitz found 7 halting TMs] that run for a large number of steps. Four of these are equivalent and were [https://discord.com/channels/960643023006490684/1331570843829932063/1337228898068463718 analyzed by Racheline] in February 2025, while the remaining three were [[User:Polygon/Page for analyses|analyzed by Polygon in October 2025.]]
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}*
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}
|<math>10 \uparrow\uparrow 9.873987</math>
|}
<nowiki>*</nowiki>equivalent to {{TM|0RB1RZ1RC_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}, {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ0RA|halt}} and {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ1RB|halt}}.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in December 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~633B(illion) TMs of which ~34.4B TMs were holdouts. Also found were ~206B halting TMs and ~392B infinite TMs. The number of holdouts was reduced to ~461M TMs (a 98.66% reduction).

Two C++ programs were run before the filters in the table.
<pre>
lr_enum 4 3 8 /dev/null /dev/null 4x3.unk.txt false
00 <= XX < 47: lr_enum_continue 4x3.in.XX 1000 /dev/null /dev/null 4x3.unk.txt.XX XX false
</pre>
Both do the initial enumeration and simple filtering. The "/dev/null" in both commands would be files where the halting and infinite TMs would be stored. The first command generates the TMs from a TNF tree for BB(4,3) of depth 8 and outputs the holdouts to 4x3.unk.txt. This file was then divided into 48 pieces, 4x3.in.XX, 0 <= XX < 47. The second commands (one for each XX) continues the enumeration by running each TM for 1,000 steps. It classifies each as halting, infinite, or unknown/holdout. Again, the halting and infinite TMs are "written" to /dev/null, i.e., they aren't saved. The holdouts are stored in 48 files: 4x3.unk.txt.XX.

For these runs the first command generated a total of ~45M TMs: ~1.86M halting, ~774K infinite, and ~42.0M holdouts. The second took the ~42.0M holdout TMs and generated a total of ~633B TMs: ~206B halting, ~392B infinite, and ~34.4B holdouts. These holdouts were used as a starting point of the filters below.

The "Description" column in the table below contain the command run. Two options are not given, "--infile=..." and an "--outfile=...". These are necessary and specify where to read and write the results, respectively. Note: The work flow was to divide the input holdouts into 48 pieces, run the command on each piece simultaneously on one of 48 cores, and then combine the 48 results into a group of holdouts.

The details are given in this table:

(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"
!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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|34,413,860,527
|30,874,934,791
|10.28%
|646.6
|1,520.36
|14,784.57
|style="text-align:left" |Reverse_Engineer_Filter.py
|rowspan="10" style="text-align:left" |[https://drive.google.com/drive/folders/1KMOVgngtUVMEA7EjxtNcsgksQ5Y4tby9?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|30,874,934,791
|12,942,386,396
|58.08%
|4,134.8
|1,204.72
|2,074.19
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|12,942,386,396
|4,534,322,415
|64.97%
|3,361.1
|694.88
|1,069.62
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|4,534,322,415
|2,959,598,830
|34.73%
|3,318.1
|131.83
|379.59
|style="text-align:left" |CPS_Filter.py --block-size=3
|-
|style="text-align:left" |Terry Ligocki
|2,959,598,830
|1,651,940,618
|44.18%
|2,700.6
|134.50
|304.42
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --no-steps --time=0.002 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|1,651,940,618
|854,984,279
|48.24%
|2,276.3
|97.25
|201.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.005 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|854,984,279
|683,163,325
|20.10%
|430.1
|110.96
|552.15
|style="text-align:left" |CPS_Filter.py --block-size=4 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|683,163,325
|460,916,384
|32.53%
|5,507.9
|11.21
|34.45
|style="text-align:left" |CPS_Filter.py --min-block-size=1 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:center" |'''Cumulative'''
|'''632,656,365,801'''
|'''460,916,384'''
|'''98.66%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

== Phase 2 ==

When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===

Starting from the results of Phase 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 33 variations were run. The holdouts were reduced from ~461B TMs to ~33.9M TMs (a 92.7% reduction). The details are given in the table below, including links to the Google Drive with the holdouts. Entries with multiple lines represent runs where all the commands in the "Description" were applied during one run.

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|460,916,384
|234,834,703
|49.05%
|96.7
|649.48
|1,324.10
|style="text-align:left" | chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 1000 NG_n 2 run
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1tFtg1eFC-AdqCzh7XNmx5O2mTQwtaNbm?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|234,834,703
|160,518,206
|31.65%
|70.9
|291.33
|920.57
|style="text-align:left" | chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 1000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|160,518,206
|132,296,033
|17.58%
|41.5
|188.86
|1,074.17
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 6 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|132,296,033
|113,193,595
|14.44%
|54.9
|96.57
|668.77
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|113,193,595
|85,920,795
|24.09%
|106.8
|70.96
|294.52
|style="text-align:left" | chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|85,920,795
|78,674,774
|8.43%
|28.9
|69.62
|825.51
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|78,674,774
|73,228,547
|6.92%
|68.7
|22.02
|318.04
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|73,228,547
|67,014,897
|8.49%
|23.2
|74.50
|878.02
|style="text-align:left" | chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|67,014,897
|57,625,231
|14.01%
|75.6
|34.49
|246.13
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|57,625,231
|48,070,606
|16.58%
|645.4
|4.11
|24.80
|style="text-align:left" | chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 30000 NG_n 10 run
|-
|style="text-align:left" |Terry Ligocki
|48,070,606
|44,254,286
|7.94%
|166.3
|6.38
|80.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 6 H 1 tH 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|44,254,286
|40,836,159
|7.72%
|188.3
|5.04
|65.29
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|40,836,159
|37,460,692
|8.27%
|192.3
|4.88
|58.99
|style="text-align:left" |
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 2 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 3 n 2 run 
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 3000 NG_n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 4 H 1 tH 1 n 2 run 
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 3 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 2 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|37,460,692
|36,167,570
|3.45%
|237.7
|1.51
|43.77
|style="text-align:left" |
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 3 H 0 tH 1 n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 10 chr_H 8 MitM_CTL NG maxT 10000 NG_n 4 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 2 run 
|-
|style="text-align:left" |Terry Ligocki
|36,167,570
|34,642,544
|4.22%
|467.2
|0.91
|21.50
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,642,544
|34,339,943
|0.87%
|383.1
|0.22
|25.12
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 0 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,339,943
|33,860,069
|1.40%
|666.5
|0.20
|14.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 2 tH 2 n 8 run
|-
|style="text-align:center" |'''Cumulative'''
|'''460,916,384'''
|'''33,860,069'''
|'''92.70%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

=== Stage 2 ===

Starting from the results of Stage 1, Terry Ligocki ran a variety of enumeration and decider codes. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~33.9M TMs to ~9.4M TMs (a 72.2% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|33,860,069
|21,065,769
|37.79%
|93.0
|38.20
|101.11
|style="text-align:left" |lr_enum_continue 4x3.in.txt 1000000 4x3.halt.txt 4x3.inf.txt 4x3.holdouts.txt 00 false
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|21,065,769
|18,949,009
|10.05%
|5,566.1
|0.11
|1.05
|style="text-align:left" |Enumerate.py max-loops 100_000 block-size 2 --tape-limit 1_000 --no-steps --time 1.0 --recursive --exp-linear-rules --lin-steps 0 --no-ctl --no-reverse-engineer --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,949,009
|18,138,027
|4.28%
|0.4
|511.59
|11953.46
|style="text-align:left" |Reverse_Engineer_Filter.py --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,138,027
|11,985,999
|33.92%
|4.8
|352.73
|1,039.95
|style="text-align:left" | chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|11,985,999
|9,988,715
|16.66%
|640.4
|0.87
|5.20
|style="text-align:left" |
chr_LRUH 24 chr_H 16 MitM_CTL NG maxT 30000 NG_n 3 run 
chr_LRUH 14 chr_H 2 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 5 run 
chr_asth 0 chr_LRUH 48 chr_H 48 MitM_CTL NG maxT 30000 NG_n 5 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 3 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 1 n 4 run 
chr_LRUH 14 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 0 n 6 run 
chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 18 chr_H 16 MitM_CTL NG maxT 30000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 1 tH 0 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 3 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|9,988,715
|9,401,447
|5.88%
|1,398.7
|0.12
|1.98
|style="text-align:left" |
chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 100000 NG_n 5 run 
chr_LRUH 22 chr_H 12 MitM_CTL NG maxT 100000 NG_n 6 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 100000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 16 H 1 tH 0 n 10 run 
chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 1000000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 6 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 2 tH 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 6 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 1 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 5 n 2 run 
|-
|style="text-align:center" |'''Cumulative'''
|'''33,860,069'''
|'''9,401,447'''
|'''72.23%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

==== Stage 3 ====
Starting from the results of Stage 2, Andrew Ducharme ran a variety of Ligocki and @mxdys deciders. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~9.4M TMs to ~5.6M (a 40.0% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:
{| 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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
| style="text-align:left" |Andrew Ducharme
|9,401,447
|7,753,702
|17.53%
|988.7
|
|
| style="text-align:left" |Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000 --block-mult=3 --time=0.5 --lin-steps=0 --no-ctl
| rowspan="28" style="text-align:left" |[https://drive.google.com/drive/folders/11FYe0CVDPczcgt4put3vsdGbeMWkM1S1?usp=sharing Google Drive]
|-
| style="text-align:left" |Andrew Ducharme
|7,753,702
|7,409,705
|4.44%
|~500
|
|
| style="text-align:left" |lr_enum_continue 10000000 (10M steps)
|-
| style="text-align:left" |Andrew Ducharme
|7,409,705
|7,192,937
|2.93%
|1858.9
|
|
| style="text-align:left" |Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000 --block-mult=12 --time=1 --tape-limit=500 --lin-steps=0 --no-ctl
|-
| style="text-align:left" |Andrew Ducharme
|7,192,937
|6,711,936
|6.69%
|3.6
|
|
| style="text-align:left" | FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 2
FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 4
|-
| style="text-align:left" |Andrew Ducharme
|6,711,936
|6,506,888
|3.05%
|~500
|
|
| style="text-align:left" |
FAR CPS_LRU maxT 100000 LRUH [1,2] remaining parameters
|-
| style="text-align:left" |Andrew Ducharme
|6,506,888
|6,298,166
|3.21%
|~2200
|
|
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH [3,4]
|-
|Andrew Ducharme
|6,298,166
|6,257,722
|0.64%
|~2000
|
|
|FAR CPS_LRU maxT 100000 LRUH 5
|-
|Andrew Ducharme
|6,257,722
|6,237,675
|0.32%
|~2400
|
|
|FAR CPS_LRU maxT 100000 LRUH 6
|-
|Andrew Ducharme
|6,237,675
|6,156,619
|1.30%
|1798.3
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=250_000 --block-mult=1 --time=1 --tape-limit=1000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|6,156,619
|6,123,679
|0.54%
|1784.3
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=250_000 --block-mult=5 --time=1 --tape-limit=1000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|6,123,679
|6,071,297
|0.86%
|~7500
|
|
|FAR CPS_LRU maxT 100000 LRUH 12
|-
|Andrew Ducharme
|6,071,297
|5,913,070
|2.61%
|~25000
|
|
|FAR CPS_LRU maxT 1000000 LRUH [1,2]
|-
|Andrew Ducharme
|5,913,070
|5,718,346
|3.29%
|15790.2
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=1_000_000 --block-mult=8 --time=10 --tape-limit=5000 --max-steps-per-macro=1_000_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|5,718,346
|5,641,006
|1.35%
|15989.4
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=10_000_000 --block-mult=3 --time=10 --tape-limit=5000 --max-steps-per-macro=1_000_000 --lin-steps=0 --no-ctl
|-
| style="text-align:center" |'''Cumulative'''
|'''9,401,447'''
|'''5,641,006'''
|'''40.00%'''
| ---
| ---
| ---
| style="text-align:left" | ---
|}

==References==

[[Category:BB Domains]][[Category:BB(4,3)]]

BB(2,6)

2026-04-26T06:52:27Z

ADucharme: /* Stage 4 */ update 2x6 progress

The 2-state, 6-symbol Busy Beaver problem, '''BB(2,6),''' is unsolved. With cryptids like [[Hydra]] in the preceding domain [[BB(2,5)]], we know that we must solve a [[Collatz-like]] problem in order to solve BB(2,6).

The current BB(2,6) champion {{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}} was discovered by Pavel Kropitz in May 2023, proving the lower bound:<math display="block">S(2,6) > \Sigma(2,6) > 10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}} > 10 \uparrow \uparrow \uparrow 3</math>

== 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 20 highest known scoring machines are:
{| class="wikitable"
|+
!TM
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|10 ↑↑↑ 3
|Pavel Kropitz
|-
|{{TM|1RB2LA1RZ1RB5RB0RB_2LA4RA3LB5LB5RA4LB|halt}}
|10 ↑↑ 19892.08
|Peacemaker II
|-
|{{TM|1RB3LA4LB0RB1RA3LA_2LA2RA4LA1RA5RB1RZ|halt}}
|10 ↑↑ 91.17
|Pavel Kropitz
|-
|{{TM|1RB2LA1RA4LA5RA0LB_1LA3RA2RB1RZ3RB4LA|halt}}
|10 ↑↑ 70.27
|Shawn Ligocki
|-
|{{TM|1RB2LB1RZ3LA2LA4RB_1LA3RB4RB1LB5LB0RA|halt}}
|10 ↑↑ 69.68
|Shawn Ligocki
|-
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|Andrew Ducharme
|-
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|Andrew Ducharme
|-
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|Andrew Ducharme
|-
|{{TM|1RB3LB3RB4LA2LA4LA_2LA2RB1LB0RA5RA1RZ|halt}}
|10 ↑↑ 21.54
|Shawn Ligocki
|-
|{{TM|1RB2LB3LA1RA0RA1RZ_1LA2RB1LB4RB5RA3LA|halt}}
|10 ↑↑ 20.58
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB1RA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB5LA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Andrew Ducharme
|-
|{{TM|1RB3RA4LB5RA5LB4RA_2LA1RZ1RB2LA5LA0LA|halt}}
|10 ↑↑ 17.08
|Andrew Ducharme
|-
|{{TM|1RB3RA4LA1LA0LA1RZ_2LA0LB1RA1LB5LB2RA|halt}}
|10 ↑↑ 15.44
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ2LA|halt}}
|10 ↑↑ 14.35
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ3RA|halt}}
|10 ↑↑ 14.17
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ1LA|halt}}
|10 ↑↑ 14.05
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ0RA|halt}}
|10 ↑↑ 13.69
|Andrew Ducharme
|-
|{{TM|1RB3LA3RA4LB2LB0LA_2LA5LB2RB0RA0RA1RZ|halt}}
|10 ↑↑ 12.42
|Andrew Ducharme
|-
|{{TM|1RB0LB4LA2RA2RB1LB_2LA4LA3LB5LA1RA1RZ|halt}}
|10 ↑↑ 11.70
|Andrew Ducharme
|}
All decimal places are truncated.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in November 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~24B(illion) TMs of which ~2.278B were holdout TMs. This was reduced to ~22M holdout TMs (a 99.02% reduction). The details are given in this table, including links to the Google Drive with the holdouts and details of the computation:

(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"
!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" |Data
|-
|style="text-align:left" |Terry Ligocki
|2,278,655,696
|2,109,114,609
|7.44%
|40.9
|1,150.90
|15,468.23
|style="text-align:left" |Reverse_Engineer_Filter.py
|style="text-align:left", rowspan="100" |[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|2,109,114,609
|683,067,538
|67.61%
|452.8
|874.77
|1,293.79
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|683,067,538
|210,993,434
|69.11%
|396.4
|330.85
|478.72
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|210,993,434
|141,680,232
|32.85%
|273.9
|70.29
|213.97
|style="text-align:left" |CPS_Filter.py --block-size=3 --max_steps=10_000
|-
|style="text-align:left" |Terry Ligocki
|141,680,232
|66,029,536
|53.40%
|486.6
|43.18
|80.87
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --time=10 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|66,029,536
|46,119,004
|30.15%
|167.4
|33.05
|109.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.01 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|46,119,004
|39,034,142
|15.36%
|170.1
|11.57
|75.34
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=12 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|39,034,142
|29,109,512
|25.43%
|2,221.6
|1.24
|4.88
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:left" |Terry Ligocki
|29,109,512
|24,536,819
|15.71%
|384.2
|3.31
|21.05
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=6 --recursive --no-steps --time=0.05 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|24,536,819
|22,302,296
|9.11%
|1,047.5
|0.59
|6.51
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=4 --recursive --no-steps --time=1.00 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|}

== Phase 2 ==
When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===
Andrew Ducharme ran another pass of "lr_enum_continue" with the maximum number of steps set to 10 million. The holdouts were reduced from ~22.3M TMs to ~20.4M TMs (a 8.72% reduction). The entry in the table below has a rather technical/arcane/cryptic description. This was an effort to capture enough information to rerun that filter in parallel with specific C++ code, lr_enum_continue, and a specific parallel queuing system, Slurm:

(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"
!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" |Data
|-
|style="text-align:left" |Andrew Ducharme
|22,302,296
|20,358,011
|8.72%
|1,350.0
|0.40
|4.59
|style="text-align:left" |lr_enum_continue ${WORK_DIR}chunk_${SLURM_ARRAY_TASK_ID} 10000000 ${WORK_DIR}halt_${SLURM_ARRAY_TASK_ID}.txt ${WORK_DIR}inf_${SLURM_ARRAY_TASK_ID}.txt ${WORK_DIR}unknown_${SLURM_ARRAY_TASK_ID}.txt "" false
|style="text-align:left" rowspan="50"|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|}

=== Stage 2 ===
Starting from the results of Stage 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 50 variations were run. The holdouts were reduced from ~20.4M TMs to ~907K TMs (a 95.5% reduction). The details are given in this table, including links to the Google Drive with the holdouts and details of the computation:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|20,358,011
|19,500,847
|4.21%
|22.0
|10.84
|257.42
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 6 run
|style="text-align:left" rowspan="50"|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|19,500,847
|18,747,861
|3.86%
|86.0
|2.43
|63.01
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|18,747,861
|4,811,076
|74.34%
|47.0
|82.33
|110.75
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|4,811,076
|2,982,075
|38.02%
|17.1
|29.74
|78.22
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|2,982,075
|2,897,340
|2.84%
|15.2
|1.55
|54.64
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 8 mod 3 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|2,897,340
|2,850,781
|1.61%
|16.7
|0.77
|48.17
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n 7 run
|-
|style="text-align:left" |Terry Ligocki
|2,850,781
|2,759,635
|3.20%
|13.7
|1.85
|58.01
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|2,759,635
|1,953,426
|29.21%
|13.6
|16.48
|56.42
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,953,426
|1,855,545
|5.01%
|2.4
|11.18
|223.14
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,855,545
|1,647,269
|11.22%
|6.6
|8.80
|78.40
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,647,269
|1,608,166
|2.37%
|3.4
|3.20
|134.96
|style="text-align:left" |chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,608,166
|1,585,745
|1.39%
|9.6
|0.65
|46.35
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|1,585,745
|1,555,673
|1.90%
|5.7
|1.47
|77.73
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 10000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,555,673
|1,428,534
|8.17%
|9.3
|3.78
|46.31
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 2 tH 0 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,428,534
|1,340,964
|6.13%
|0.8
|29.70
|484.55
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 10000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,340,964
|1,286,439
|4.07%
|0.8
|18.40
|452.56
|style="text-align:left" |chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,286,439
|1,273,911
|0.97%
|0.8
|4.20
|430.88
|style="text-align:left" |chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,273,911
|1,265,198
|0.68%
|0.8
|2.88
|420.73
|style="text-align:left" |chr_LRUH 3 chr_H 1 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,265,198
|1,258,925
|0.50%
|0.9
|1.99
|400.83
|style="text-align:left" |chr_LRUH 8 chr_H 6 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,258,925
|1,242,136
|1.33%
|0.8
|5.51
|412.84
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 0 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,242,136
|1,231,731
|0.84%
|1.0
|2.78
|331.77
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 2 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,231,731
|1,216,646
|1.22%
|1.0
|4.15
|338.72
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 12 H 0 tH 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,216,646
|1,214,294
|0.19%
|0.9
|0.76
|393.03
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 2 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,214,294
|1,213,431
|0.07%
|0.9
|0.28
|391.30
|style="text-align:left" |chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,213,431
|1,211,390
|0.17%
|1.1
|0.52
|307.13
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 1 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,211,390
|1,209,989
|0.12%
|1.1
|0.35
|306.09
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,209,989
|1,209,974
|0.00%
|0.9
|0.00
|381.42
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 16 H 1 tH 0 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,209,974
|1,201,890
|0.67%
|2.5
|0.90
|134.19
|style="text-align:left" |chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,201,890
|1,200,086
|0.15%
|1.3
|0.37
|248.36
|style="text-align:left" |chr_LRUH 10 chr_H 6 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,200,086
|1,199,734
|0.03%
|1.2
|0.08
|270.32
|style="text-align:left" |chr_asth 0 chr_LRUH 3 chr_H 3 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,199,734
|1,198,893
|0.07%
|2.3
|0.10
|147.66
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 2 mod 6 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,198,893
|1,165,493
|2.79%
|4.5
|2.05
|73.44
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 4 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,165,493
|1,153,863
|1.00%
|9.3
|0.35
|34.88
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 0 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,153,863
|1,144,711
|0.79%
|3.7
|0.69
|87.51
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 5 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,144,711
|1,127,789
|1.48%
|7.9
|0.60
|40.26
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 30000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,127,789
|1,124,762
|0.27%
|4.7
|0.18
|66.75
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 0 tH 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|1,124,762
|1,117,226
|0.67%
|5.6
|0.37
|55.36
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 12 H 0 tH 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,117,226
|1,109,057
|0.73%
|7.7
|0.30
|40.49
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,109,057
|1,083,097
|2.34%
|11.4
|0.63
|27.06
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 30000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,083,097
|1,077,833
|0.49%
|11.2
|0.13
|26.81
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 100000 NG_n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,077,833
|1,066,795
|1.02%
|24.1
|0.13
|12.40
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 2 tH 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,066,795
|1,039,229
|2.58%
|52.6
|0.15
|5.64
|style="text-align:left" |chr_LRUH 14 chr_H 6 MitM_CTL NG maxT 100000 NG_n 11 run
|-
|style="text-align:left" |Terry Ligocki
|1,039,229
|1,019,286
|1.92%
|43.5
|0.13
|6.63
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 12 mod 1 n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,019,286
|993,556
|2.52%
|66.8
|0.11
|4.24
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 8 H 2 tH 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|993,556
|985,718
|0.79%
|78.3
|0.03
|3.53
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 1 tH 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|985,718
|981,095
|0.47%
|83.7
|0.02
|3.27
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 1 tH 0 n 9 run
|-
|style="text-align:left" |Terry Ligocki
|981,095
|975,912
|0.53%
|79.4
|0.02
|3.43
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 16 mod 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|975,912
|974,180
|0.18%
|84.6
|0.01
|3.20
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 16 mod 4 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|974,180
|971,254
|0.30%
|96.9
|0.01
|2.79
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 12 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|971,254
|970,101
|0.12%
|105.6
|0.00
|2.56
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 0 tH 0 n 18 run
|}

=== Stage 3 ===
Starting from the results of Stage 2, Andrew Ducharme ran "lr_enum_continue" with the maximum number of steps set to 100 million, then "Enumerate.py" with various parameters. A total of 10 Enumerate variations were run. The holdouts were reduced from ~970K TMs to ~867K TMs (a 10.63% reduction). The details are given in this table, including links to the Google Drive with the holdouts and details of the computation:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Andrew Ducharme
|970,101
|939,447
|3.16%
| --
| --
| --
|style="text-align:left" |lr_enum_continue 100_000_000 steps
| rowspan="11" |[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|-
| style="text-align:left" |Andrew Ducharme
|939,447
|903,224
|3.86%
|440.3
|0.03
|0.59
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=4 --no-ctl --lin-steps=0 --time=2 --force --save-freq=1000
|-
| style="text-align:left" |Andrew Ducharme
|903,224
|895,813
|0.82%
|647.7
|0.00
|0.39
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=3 --no-ctl --lin-steps=0 --time=3 --force --save-freq=1000
|-
| style="text-align:left" |Andrew Ducharme
|895,813
|889,838
|0.67%
|609.3
|0.00
|0.41
| style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=8 --no-ctl --lin-steps=0 --time=4 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|889,838
|880,278
|1.07%
|1,638.9
|0.00
|0.15
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=12 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|880,278
|877,485
|0.32%
|1,885.5
|0.00
|0.13
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=6 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|877,485
|875,062
|0.28%
|2,068.8
|0.00
|0.12
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=5 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|875,062
|873,469
|0.18%
|1,785.4
|0.00
|0.14
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=7 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|873,469
|870,085
|0.39%
|9,270.0
|0.00
|0.03
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=2 --tape-limit=500 --time=120 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|870,085
|869,001
|0.12%
|4,498.3
|0.00
|0.05
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=10_000_000 --block-mult=60 --tape-limit=5000 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|869,001
|867,008
|0.23%
|3997.4
|0.00
|0.06
|style="text-align:left"|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=9 --tape-limit=5000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl --force --save-freq=250
|}
The total time spent on the lr_enum_continue computation was not recorded.

=== Stage 4 ===
Following the release of @mxdys's implementation of FAR deciders in C++, these deciders were applied to the 2x6 holdouts by Andrew Ducharme. The details are given in this table, including links to the Google Drive with the holdouts and solved TMs per decider:

(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"
! colspan="2" |Holdout TMs
! rowspan="2" |<math>*^1</math>
! rowspan="2" |<math>*^2</math>
! colspan="2" |TMs/sec/core
! rowspan="2" |Description
! rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|867,008
|811,301
|6.43%
|0.043
|364.10
|5,666.72
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 2 H 1 tH 1 n 2
| rowspan="28" |[https://drive.google.com/drive/folders/18njhmOzRc67zCmVuLd0aDxl6ETBhL1gy?usp=sharing Google Drive]
|-
|811,301
|806,119
|0.64%
|0.159
|9.03
|1,413.42
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 3 H 1 tH 1 n 2
|-
|806,119
|736,690
|8.61%
|0.548
|35.21
|408.78
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 1 tH 1 n 2
|-
|736,690
|736,504
|0.03%
|0.009
|5.81
|23,021.56
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 1 H 1 tH 1 n 1
|-
|736,504
|735,317
|0.16%
|0.058
|5.71
|3,540.88
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 2
|-
|735,317
|733,717
|0.22%
|0.341
|1.30
|599.28
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 2 tH 2 n 2
|-
|733,717
|673,920
|8.15%
|3.8
|4.43
|54.32
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 2 tH 2 n 4
|-
|673,920
|652,828
|3.13%
|~10
| ---
| ---
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 6 H 2 tH 2 n 4
|-
|652,828
|645,264
|1.16%
|~12
| ---
| ---
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 8 H 2 tH 2 n 4
|-
|645,264
|641,388
|0.60%
|~15
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 100000 LRUH 10 H 2 tH 2 n 10
|-
|641,388
|635,505
|0.92%
|~200
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 10 H 1 tH 2 n 10
|-
|635,505
|616,639
|2.97%
| ---
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 2 H 0 tH 0 n [3-10]
|-
|616,639
|592,039
|3.99%
|~700
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 3 H 0 tH 0 n [1-10]
|-
|592,039
|576,938
|2.55%
|~800
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 3 H [0-1] tH [0-1] n [1-10]
|-
|576,938
|572,963
|0.69%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4 H 0 tH 0 n [1-10]
|-
|572,963
|567,971
|0.87%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4 H 2 tH 0 n [1-10]
|-
|567,971
|566,096
|0.33%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 6 H 0 tH 0 n [1-10]
|-
|566,096
|564,290
|0.32%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 0 tH [0,2] n [1-10]
|-
|564,290
|559,553
|0.84%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 2 tH 1 n [1-10]
|-
|559,553
|558,039
|0.27%
|~900
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 2 tH 2 n [1-10]
|-
|558,039
|556,814
|0.22%
|~14,000
| ---
| ---
|FAR CPS_LRU maxT 1000000 LRUH [12,16] H [0-2] tH [0-2] n [1-10]
|-
|556,814
|554,479
|0.42%
|~3,600
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH [1-3]
|-
|554,479
|551,586
|0.52%
|~5000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4
|-
|551,586
|548,993
|0.47%
|~13,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 5
|-
|548,993
|545,005
|0.73%
|~57,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 6 and 8
|-
|545,005
|542,325
|0.49%
|6851.2
|0.00
|0.022
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=96 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|-
|542,325
|537,393
|0.91%
|9032.1
|0.00
|0.017
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=2 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|-
|537,393
|536,112
|0.24%
|8969.4
|0.00
|0.017
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=3 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|}

== References ==


[[Category: BB Domains]][[Category:BB(2,6)]]

BB(4,3)

2026-04-26T05:24:49Z

ADucharme: /* Top Halters */ spaces

The Busy Beaver problem for 4 states and 3 symbols is unsolved. The existence of [[Cryptids]] in the domain is given by the discovery of [[Bigfoot]] in [[BB(3,3)]]. The current [[Champions#3-Symbol TMs|champion]] is {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} which was discovered by Pavel Kropitz in May 2024 along with 6 other long running machines. It was [[User:Polygon/Page for analyses#1RB1RD1LC 2LB1RB1LC 1RZ1LA1LD 0RB2RA2RD (bbch)|analyzed by Polygon]] in Oct 2025, demonstrating the lower bounds:

<math display="block">S(4,3) > \Sigma(4,3) > 10 \uparrow^{4} 4</math>

== Top Halters ==
The longest running halting BB(4,3) TMs are split amongst two classes: the pentational and hexational TMs found by Pavel Kropitz outlined in the Potential Champions section, and the tetrational TMs found by comprehensive holdout filtering by Terry Ligocki. 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 longest running halters found by Pavel Kropitz are:
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|}
The top 20 scoring halting machines found by comprehensive search are:
{| class="wikitable"
|+
!Standard format
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|~10 ↑↑ 1023.47221
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 619.07737
|Andrew Ducharme
|-
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|~10 ↑↑ 512.10945
|Andrew Ducharme
|-
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|~10 ↑↑ 439.02781
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 234.06408
|Andrew Ducharme
|-
|{{TM|1RB0LC1RC_1LA2RB1LB_1RC2LA0RD_2LB1RZ2LC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LA1RA_1LA0RC1LC_1LC2RB0LD_2RA1RZ2RC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_2RC1LB2RD_1LD2LA0LB_0LA1RZ0LC|halt}}
|~10 ↑↑ 178.48320
|Andrew Ducharme
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA1LB_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB1LD0RC_2LC0RB1RA_1RA0LB1RD_0LA2LA1RZ|halt}}
|~10 ↑↑ 158.81916
|Andrew Ducharme
|-
|{{TM|1RB1RC1RB_1LC0RA2LD_2RA0LD1RZ_0LB2LD1RD|halt}}
|~10 ↑↑ 154.52968
|Andrew Ducharme
|-
|{{TM|1RB1LA1RD_2LA0LC2LD_1RZ2RA2LB_0LC2RC1RA|halt}}
|~10 ↑↑ 147.26175
|Andrew Ducharme
|-
|{{TM|1RB0RB1LC_2LC0LD1RA_2RB2LD1RZ_2LA2LB0LD|halt}}
|~10 ↑↑ 141.44248
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ2LD1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC1LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA---|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|}

== Potential Champions ==
In May 2024, [https://discord.com/channels/960643023006490684/1026577255754903572/1243253180297646120 Pavel Kropitz found 7 halting TMs] that run for a large number of steps. Four of these are equivalent and were [https://discord.com/channels/960643023006490684/1331570843829932063/1337228898068463718 analyzed by Racheline] in February 2025, while the remaining three were [[User:Polygon/Page for analyses|analyzed by Polygon in October 2025.]]
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}*
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}
|<math>10 \uparrow\uparrow 9.873987</math>
|}
<nowiki>*</nowiki>equivalent to {{TM|0RB1RZ1RC_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}, {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ0RA|halt}} and {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ1RB|halt}}.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in December 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~633B(illion) TMs of which ~34.4B TMs were holdouts. Also found were ~206B halting TMs and ~392B infinite TMs. The number of holdouts was reduced to ~461M TMs (a 98.66% reduction).

Two C++ programs were run before the filters in the table.
<pre>
lr_enum 4 3 8 /dev/null /dev/null 4x3.unk.txt false
00 <= XX < 47: lr_enum_continue 4x3.in.XX 1000 /dev/null /dev/null 4x3.unk.txt.XX XX false
</pre>
Both do the initial enumeration and simple filtering. The "/dev/null" in both commands would be files where the halting and infinite TMs would be stored. The first command generates the TMs from a TNF tree for BB(4,3) of depth 8 and outputs the holdouts to 4x3.unk.txt. This file was then divided into 48 pieces, 4x3.in.XX, 0 <= XX < 47. The second commands (one for each XX) continues the enumeration by running each TM for 1,000 steps. It classifies each as halting, infinite, or unknown/holdout. Again, the halting and infinite TMs are "written" to /dev/null, i.e., they aren't saved. The holdouts are stored in 48 files: 4x3.unk.txt.XX.

For these runs the first command generated a total of ~45M TMs: ~1.86M halting, ~774K infinite, and ~42.0M holdouts. The second took the ~42.0M holdout TMs and generated a total of ~633B TMs: ~206B halting, ~392B infinite, and ~34.4B holdouts. These holdouts were used as a starting point of the filters below.

The "Description" column in the table below contain the command run. Two options are not given, "--infile=..." and an "--outfile=...". These are necessary and specify where to read and write the results, respectively. Note: The work flow was to divide the input holdouts into 48 pieces, run the command on each piece simultaneously on one of 48 cores, and then combine the 48 results into a group of holdouts.

The details are given in this table:

(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"
!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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|34,413,860,527
|30,874,934,791
|10.28%
|646.6
|1,520.36
|14,784.57
|style="text-align:left" |Reverse_Engineer_Filter.py
|rowspan="10" style="text-align:left" |[https://drive.google.com/drive/folders/1KMOVgngtUVMEA7EjxtNcsgksQ5Y4tby9?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|30,874,934,791
|12,942,386,396
|58.08%
|4,134.8
|1,204.72
|2,074.19
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|12,942,386,396
|4,534,322,415
|64.97%
|3,361.1
|694.88
|1,069.62
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|4,534,322,415
|2,959,598,830
|34.73%
|3,318.1
|131.83
|379.59
|style="text-align:left" |CPS_Filter.py --block-size=3
|-
|style="text-align:left" |Terry Ligocki
|2,959,598,830
|1,651,940,618
|44.18%
|2,700.6
|134.50
|304.42
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --no-steps --time=0.002 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|1,651,940,618
|854,984,279
|48.24%
|2,276.3
|97.25
|201.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.005 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|854,984,279
|683,163,325
|20.10%
|430.1
|110.96
|552.15
|style="text-align:left" |CPS_Filter.py --block-size=4 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|683,163,325
|460,916,384
|32.53%
|5,507.9
|11.21
|34.45
|style="text-align:left" |CPS_Filter.py --min-block-size=1 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:center" |'''Cumulative'''
|'''632,656,365,801'''
|'''460,916,384'''
|'''98.66%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

== Phase 2 ==

When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===

Starting from the results of Phase 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 33 variations were run. The holdouts were reduced from ~461B TMs to ~33.9M TMs (a 92.7% reduction). The details are given in the table below, including links to the Google Drive with the holdouts. Entries with multiple lines represent runs where all the commands in the "Description" were applied during one run.

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|460,916,384
|234,834,703
|49.05%
|96.7
|649.48
|1,324.10
|style="text-align:left" | chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 1000 NG_n 2 run
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1tFtg1eFC-AdqCzh7XNmx5O2mTQwtaNbm?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|234,834,703
|160,518,206
|31.65%
|70.9
|291.33
|920.57
|style="text-align:left" | chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 1000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|160,518,206
|132,296,033
|17.58%
|41.5
|188.86
|1,074.17
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 6 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|132,296,033
|113,193,595
|14.44%
|54.9
|96.57
|668.77
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|113,193,595
|85,920,795
|24.09%
|106.8
|70.96
|294.52
|style="text-align:left" | chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|85,920,795
|78,674,774
|8.43%
|28.9
|69.62
|825.51
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|78,674,774
|73,228,547
|6.92%
|68.7
|22.02
|318.04
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|73,228,547
|67,014,897
|8.49%
|23.2
|74.50
|878.02
|style="text-align:left" | chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|67,014,897
|57,625,231
|14.01%
|75.6
|34.49
|246.13
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|57,625,231
|48,070,606
|16.58%
|645.4
|4.11
|24.80
|style="text-align:left" | chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 30000 NG_n 10 run
|-
|style="text-align:left" |Terry Ligocki
|48,070,606
|44,254,286
|7.94%
|166.3
|6.38
|80.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 6 H 1 tH 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|44,254,286
|40,836,159
|7.72%
|188.3
|5.04
|65.29
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|40,836,159
|37,460,692
|8.27%
|192.3
|4.88
|58.99
|style="text-align:left" |
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 2 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 3 n 2 run 
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 3000 NG_n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 4 H 1 tH 1 n 2 run 
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 3 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 2 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|37,460,692
|36,167,570
|3.45%
|237.7
|1.51
|43.77
|style="text-align:left" |
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 3 H 0 tH 1 n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 10 chr_H 8 MitM_CTL NG maxT 10000 NG_n 4 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 2 run 
|-
|style="text-align:left" |Terry Ligocki
|36,167,570
|34,642,544
|4.22%
|467.2
|0.91
|21.50
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,642,544
|34,339,943
|0.87%
|383.1
|0.22
|25.12
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 0 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,339,943
|33,860,069
|1.40%
|666.5
|0.20
|14.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 2 tH 2 n 8 run
|-
|style="text-align:center" |'''Cumulative'''
|'''460,916,384'''
|'''33,860,069'''
|'''92.70%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

=== Stage 2 ===

Starting from the results of Stage 1, Terry Ligocki ran a variety of enumeration and decider codes. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~33.9M TMs to ~9.4M TMs (a 72.2% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|33,860,069
|21,065,769
|37.79%
|93.0
|38.20
|101.11
|style="text-align:left" |lr_enum_continue 4x3.in.txt 1000000 4x3.halt.txt 4x3.inf.txt 4x3.holdouts.txt 00 false
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|21,065,769
|18,949,009
|10.05%
|5,566.1
|0.11
|1.05
|style="text-align:left" |Enumerate.py max-loops 100_000 block-size 2 --tape-limit 1_000 --no-steps --time 1.0 --recursive --exp-linear-rules --lin-steps 0 --no-ctl --no-reverse-engineer --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,949,009
|18,138,027
|4.28%
|0.4
|511.59
|11953.46
|style="text-align:left" |Reverse_Engineer_Filter.py --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,138,027
|11,985,999
|33.92%
|4.8
|352.73
|1,039.95
|style="text-align:left" | chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|11,985,999
|9,988,715
|16.66%
|640.4
|0.87
|5.20
|style="text-align:left" |
chr_LRUH 24 chr_H 16 MitM_CTL NG maxT 30000 NG_n 3 run 
chr_LRUH 14 chr_H 2 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 5 run 
chr_asth 0 chr_LRUH 48 chr_H 48 MitM_CTL NG maxT 30000 NG_n 5 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 3 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 1 n 4 run 
chr_LRUH 14 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 0 n 6 run 
chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 18 chr_H 16 MitM_CTL NG maxT 30000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 1 tH 0 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 3 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|9,988,715
|9,401,447
|5.88%
|1,398.7
|0.12
|1.98
|style="text-align:left" |
chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 100000 NG_n 5 run 
chr_LRUH 22 chr_H 12 MitM_CTL NG maxT 100000 NG_n 6 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 100000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 16 H 1 tH 0 n 10 run 
chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 1000000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 6 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 2 tH 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 6 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 1 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 5 n 2 run 
|-
|style="text-align:center" |'''Cumulative'''
|'''33,860,069'''
|'''9,401,447'''
|'''72.23%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

==== Stage 3 ====
Starting from the results of Stage 2, Andrew Ducharme ran a variety of Ligocki and @mxdys deciders. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~9.4M TMs to ~5.6M (a 40.0% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:
{| 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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
| style="text-align:left" |Andrew Ducharme
|9,401,447
|7,753,702
|17.53%
|988.7
|
|
| style="text-align:left" |Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000 --block-mult=3 --time=0.5 --lin-steps=0 --no-ctl
| rowspan="28" style="text-align:left" |[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_?usp=drive_link Google Drive]
|-
| style="text-align:left" |Andrew Ducharme
|7,753,702
|7,409,705
|4.44%
|~500
|
|
| style="text-align:left" |lr_enum_continue 10000000 (10M steps)
|-
| style="text-align:left" |Andrew Ducharme
|7,409,705
|7,192,937
|2.93%
|1858.9
|
|
| style="text-align:left" |Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000 --block-mult=12 --time=1 --tape-limit=500 --lin-steps=0 --no-ctl
|-
| style="text-align:left" |Andrew Ducharme
|7,192,937
|6,711,936
|6.69%
|3.6
|
|
| style="text-align:left" | FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 2
FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 4
|-
| style="text-align:left" |Andrew Ducharme
|6,711,936
|6,506,888
|3.05%
|~500
|
|
| style="text-align:left" |
FAR CPS_LRU maxT 100000 LRUH [1,2] remaining parameters
|-
| style="text-align:left" |Andrew Ducharme
|6,506,888
|6,298,166
|3.21%
|~2200
|
|
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH [3,4]
|-
|Andrew Ducharme
|6,298,166
|6,257,722
|0.64%
|~2000
|
|
|FAR CPS_LRU maxT 100000 LRUH 5
|-
|Andrew Ducharme
|6,257,722
|6,237,675
|0.32%
|~2400
|
|
|FAR CPS_LRU maxT 100000 LRUH 6
|-
|Andrew Ducharme
|6,237,675
|6,156,619
|1.30%
|1798.3
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=250_000 --block-mult=1 --time=1 --tape-limit=1000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|6,156,619
|6,123,679
|0.54%
|1784.3
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=250_000 --block-mult=5 --time=1 --tape-limit=1000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|6,123,679
|6,071,297
|0.86%
|~7500
|
|
|FAR CPS_LRU maxT 100000 LRUH 12
|-
|Andrew Ducharme
|6,071,297
|5,913,070
|2.61%
|~25000
|
|
|FAR CPS_LRU maxT 1000000 LRUH [1,2]
|-
|Andrew Ducharme
|5,913,070
|5,718,346
|3.29%
|15790.2
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=1_000_000 --block-mult=8 --time=10 --tape-limit=5000 --max-steps-per-macro=1_000_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|5,718,346
|5,641,006
|1.35%
|15989.4
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=10_000_000 --block-mult=3 --time=10 --tape-limit=5000 --max-steps-per-macro=1_000_000 --lin-steps=0 --no-ctl
|-
| style="text-align:center" |'''Cumulative'''
|'''9,401,447'''
|'''5,641,006'''
|'''40.00%'''
| ---
| ---
| ---
| style="text-align:left" | ---
|}

==References==

[[Category:BB Domains]][[Category:BB(4,3)]]

User:ADucharme

2026-04-25T23:01:40Z

ADucharme:

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts remaining from that initial push. Specifically, my enumeration work has solved

* 67.27% of the last 86,129,304, and 12.65% of the last 20,405,295, BB(7) holdouts
* 44.74% of the last 970,101 BB(2,6) holdouts
* 40.00% of the last 9,401,447 BB(4,3) holdouts

I've also tried pen-and-paper analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively-studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last ~1500 BB(6) holdouts, I solved 67 and counting. Partial credit for some of these machines goes to Peacemaker II, who identifies permutations of machines I solved in the holdout list. Because of the shared behavior between permutations, I can apply the decider which solved to the original TM I found to the permutations, and often solve permutations too.

Solved halting TMs (49) with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs (18) with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9
1RB0RB_1LC1RA_0LA1RD_1LA1LE_1LF1LD_---0LC FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 3 n 12
1RB0LD_1RC0RE_0LA0RC_1LA1LD_0RF1RA_---1RC FAR CPS_LRU maxT 10000000 LRUH 6 H 3 tH 0 n 9
1RB1LB_1LC1RE_0RD0LB_0LB1RA_1LA0RF_---0RC FAR CPS_LRU maxT 10000000 LRUH 7 H 3 tH 1 n 4
1RB0LD_0RC1RF_1RD0RA_1LE1RB_1LC0LE_1RC--- FAR CPS_LRU maxT 10000000 LRUH 7 H 4 tH 1 n 24
1RB0LA_0RC---_1RD1RE_1LA1LD_1RD0RF_0RC1RC FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB0LA_1RC1RA_0LD1LA_1LF1RE_0RD0RE_0LC--- FAR RWL_mod maxT 10000000 H 4 mod 1 n 8
1RB1RF_1LC1LB_---0LD_1RE0LD_0RA1RA_0LE0RE FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB1RD_0RC1RE_1LD0RE_1LB---_0RA1LF_0LE0LF FAR CPS_LRU maxT 1000000 LRUH 32 H 1 tH 29 n 12
1RB1RE_1LC0RF_1RE0LD_1LC0LB_1RA0RE_1RC--- FAR CPS_LRU maxT 1000000 LRUH 32 H 4 tH 20 n 24
1RB0RE_1LC1RA_0LA1LD_1RE1LC_0RF1RB_---0LC FAR CPS_LRU maxT 1000000 LRUH 17 H 4 tH 13 n 3

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Games ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain. These remain my best-ever entries in these particular Busy Beaver games.

==== BB(7) ====
{| class="wikitable sortable"
|Place
|TM
|Score
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|5
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG}}
|10 ↑↑ 403.84
|-
|9
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}}
|10 ↑↑ 243.88
|}

{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}} was a bit of co-discovery: Iijil first enumerated the TM and I first showed it was halting.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}

==== BB(4,3) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|4
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|~10 ↑↑ 1023.47221
|-
|5
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 619.07737
|-
|6
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|~10 ↑↑ 512.10945
|-
|7
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|~10 ↑↑ 439.02781
|-
|8
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 234.06408
|}

User:ADucharme

2026-04-25T23:00:36Z

ADucharme: summary

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts since that enumeration. Specifically, my enumeration work has solved

* 67.27% of the last 86,129,304, and 12.65% of the last 20,405,295, BB(7) holdouts
* 40.00% of the last 9,401,447 BB(4,3) holdouts
* 44.74% of the last 970,101 BB(2,6) holdouts

I've also tried pen-and-paper analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last ~1500 BB(6) holdouts, I solved 67 and counting. Partial credit for some of these machines goes to Peacemaker II, who identifies permutations of machines I solved in the holdout list. Because of the shared behavior between permutations, I can apply the decider which solved to the original TM I found to the permutations, and often solve permutations too.

Solved halting TMs (49) with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs (18) with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9
1RB0RB_1LC1RA_0LA1RD_1LA1LE_1LF1LD_---0LC FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 3 n 12
1RB0LD_1RC0RE_0LA0RC_1LA1LD_0RF1RA_---1RC FAR CPS_LRU maxT 10000000 LRUH 6 H 3 tH 0 n 9
1RB1LB_1LC1RE_0RD0LB_0LB1RA_1LA0RF_---0RC FAR CPS_LRU maxT 10000000 LRUH 7 H 3 tH 1 n 4
1RB0LD_0RC1RF_1RD0RA_1LE1RB_1LC0LE_1RC--- FAR CPS_LRU maxT 10000000 LRUH 7 H 4 tH 1 n 24
1RB0LA_0RC---_1RD1RE_1LA1LD_1RD0RF_0RC1RC FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB0LA_1RC1RA_0LD1LA_1LF1RE_0RD0RE_0LC--- FAR RWL_mod maxT 10000000 H 4 mod 1 n 8
1RB1RF_1LC1LB_---0LD_1RE0LD_0RA1RA_0LE0RE FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB1RD_0RC1RE_1LD0RE_1LB---_0RA1LF_0LE0LF FAR CPS_LRU maxT 1000000 LRUH 32 H 1 tH 29 n 12
1RB1RE_1LC0RF_1RE0LD_1LC0LB_1RA0RE_1RC--- FAR CPS_LRU maxT 1000000 LRUH 32 H 4 tH 20 n 24
1RB0RE_1LC1RA_0LA1LD_1RE1LC_0RF1RB_---0LC FAR CPS_LRU maxT 1000000 LRUH 17 H 4 tH 13 n 3

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Games ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain. These remain my best-ever entries in these particular Busy Beaver games.

==== BB(7) ====
{| class="wikitable sortable"
|Place
|TM
|Score
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|5
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG}}
|10 ↑↑ 403.84
|-
|9
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}}
|10 ↑↑ 243.88
|}

{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}} was a bit of co-discovery: Iijil first enumerated the TM and I first showed it was halting.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}

==== BB(4,3) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|4
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|~10 ↑↑ 1023.47221
|-
|5
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 619.07737
|-
|6
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|~10 ↑↑ 512.10945
|-
|7
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|~10 ↑↑ 439.02781
|-
|8
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 234.06408
|}

BB(4,3)

2026-04-25T22:18:42Z

ADucharme: /* Phase 2 */ Phase 2, Stage 3 table added

The Busy Beaver problem for 4 states and 3 symbols is unsolved. The existence of [[Cryptids]] in the domain is given by the discovery of [[Bigfoot]] in [[BB(3,3)]]. The current [[Champions#3-Symbol TMs|champion]] is {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} which was discovered by Pavel Kropitz in May 2024 along with 6 other long running machines. It was [[User:Polygon/Page for analyses#1RB1RD1LC 2LB1RB1LC 1RZ1LA1LD 0RB2RA2RD (bbch)|analyzed by Polygon]] in Oct 2025, demonstrating the lower bounds:

<math display="block">S(4,3) > \Sigma(4,3) > 10 \uparrow^{4} 4</math>

== Top Halters ==
The longest running halting BB(4,3) TMs are split amongst two classes: the pentational and hexational TMs found by Pavel Kropitz outlined in the Potential Champions section, and the tetrational TMs found by comprehensive holdout filtering by Terry Ligocki. 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 longest running halters found by Pavel Kropitz are:
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|}
The top 20 scoring halting machines found by comprehensive search are:
{| class="wikitable"
|+
!Standard format
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|~10↑↑1023.47221
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10↑↑619.07737
|Andrew Ducharme
|-
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|~10↑↑512.10945
|Andrew Ducharme
|-
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|~10↑↑439.02781
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10↑↑234.06408
|Andrew Ducharme
|-
|{{TM|1RB0LC1RC_1LA2RB1LB_1RC2LA0RD_2LB1RZ2LC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LA1RA_1LA0RC1LC_1LC2RB0LD_2RA1RZ2RC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_2RC1LB2RD_1LD2LA0LB_0LA1RZ0LC|halt}}
|~10 ↑↑ 178.48320
|Andrew Ducharme
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA1LB_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB1LD0RC_2LC0RB1RA_1RA0LB1RD_0LA2LA1RZ|halt}}
|~10 ↑↑ 158.81916
|Andrew Ducharme
|-
|{{TM|1RB1RC1RB_1LC0RA2LD_2RA0LD1RZ_0LB2LD1RD|halt}}
|~10 ↑↑ 154.52968
|Andrew Ducharme
|-
|{{TM|1RB1LA1RD_2LA0LC2LD_1RZ2RA2LB_0LC2RC1RA|halt}}
|~10 ↑↑ 147.26175
|Andrew Ducharme
|-
|{{TM|1RB0RB1LC_2LC0LD1RA_2RB2LD1RZ_2LA2LB0LD|halt}}
|~10 ↑↑ 141.44248
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ2LD1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC1LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA---|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|}

== Potential Champions ==
In May 2024, [https://discord.com/channels/960643023006490684/1026577255754903572/1243253180297646120 Pavel Kropitz found 7 halting TMs] that run for a large number of steps. Four of these are equivalent and were [https://discord.com/channels/960643023006490684/1331570843829932063/1337228898068463718 analyzed by Racheline] in February 2025, while the remaining three were [[User:Polygon/Page for analyses|analyzed by Polygon in October 2025.]]
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}*
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}
|<math>10 \uparrow\uparrow 9.873987</math>
|}
<nowiki>*</nowiki>equivalent to {{TM|0RB1RZ1RC_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}, {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ0RA|halt}} and {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ1RB|halt}}.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in December 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~633B(illion) TMs of which ~34.4B TMs were holdouts. Also found were ~206B halting TMs and ~392B infinite TMs. The number of holdouts was reduced to ~461M TMs (a 98.66% reduction).

Two C++ programs were run before the filters in the table.
<pre>
lr_enum 4 3 8 /dev/null /dev/null 4x3.unk.txt false
00 <= XX < 47: lr_enum_continue 4x3.in.XX 1000 /dev/null /dev/null 4x3.unk.txt.XX XX false
</pre>
Both do the initial enumeration and simple filtering. The "/dev/null" in both commands would be files where the halting and infinite TMs would be stored. The first command generates the TMs from a TNF tree for BB(4,3) of depth 8 and outputs the holdouts to 4x3.unk.txt. This file was then divided into 48 pieces, 4x3.in.XX, 0 <= XX < 47. The second commands (one for each XX) continues the enumeration by running each TM for 1,000 steps. It classifies each as halting, infinite, or unknown/holdout. Again, the halting and infinite TMs are "written" to /dev/null, i.e., they aren't saved. The holdouts are stored in 48 files: 4x3.unk.txt.XX.

For these runs the first command generated a total of ~45M TMs: ~1.86M halting, ~774K infinite, and ~42.0M holdouts. The second took the ~42.0M holdout TMs and generated a total of ~633B TMs: ~206B halting, ~392B infinite, and ~34.4B holdouts. These holdouts were used as a starting point of the filters below.

The "Description" column in the table below contain the command run. Two options are not given, "--infile=..." and an "--outfile=...". These are necessary and specify where to read and write the results, respectively. Note: The work flow was to divide the input holdouts into 48 pieces, run the command on each piece simultaneously on one of 48 cores, and then combine the 48 results into a group of holdouts.

The details are given in this table:

(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"
!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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|34,413,860,527
|30,874,934,791
|10.28%
|646.6
|1,520.36
|14,784.57
|style="text-align:left" |Reverse_Engineer_Filter.py
|rowspan="10" style="text-align:left" |[https://drive.google.com/drive/folders/1KMOVgngtUVMEA7EjxtNcsgksQ5Y4tby9?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|30,874,934,791
|12,942,386,396
|58.08%
|4,134.8
|1,204.72
|2,074.19
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|12,942,386,396
|4,534,322,415
|64.97%
|3,361.1
|694.88
|1,069.62
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|4,534,322,415
|2,959,598,830
|34.73%
|3,318.1
|131.83
|379.59
|style="text-align:left" |CPS_Filter.py --block-size=3
|-
|style="text-align:left" |Terry Ligocki
|2,959,598,830
|1,651,940,618
|44.18%
|2,700.6
|134.50
|304.42
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --no-steps --time=0.002 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|1,651,940,618
|854,984,279
|48.24%
|2,276.3
|97.25
|201.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.005 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|854,984,279
|683,163,325
|20.10%
|430.1
|110.96
|552.15
|style="text-align:left" |CPS_Filter.py --block-size=4 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|683,163,325
|460,916,384
|32.53%
|5,507.9
|11.21
|34.45
|style="text-align:left" |CPS_Filter.py --min-block-size=1 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:center" |'''Cumulative'''
|'''632,656,365,801'''
|'''460,916,384'''
|'''98.66%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

== Phase 2 ==

When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===

Starting from the results of Phase 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 33 variations were run. The holdouts were reduced from ~461B TMs to ~33.9M TMs (a 92.7% reduction). The details are given in the table below, including links to the Google Drive with the holdouts. Entries with multiple lines represent runs where all the commands in the "Description" were applied during one run.

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|460,916,384
|234,834,703
|49.05%
|96.7
|649.48
|1,324.10
|style="text-align:left" | chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 1000 NG_n 2 run
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1tFtg1eFC-AdqCzh7XNmx5O2mTQwtaNbm?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|234,834,703
|160,518,206
|31.65%
|70.9
|291.33
|920.57
|style="text-align:left" | chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 1000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|160,518,206
|132,296,033
|17.58%
|41.5
|188.86
|1,074.17
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 6 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|132,296,033
|113,193,595
|14.44%
|54.9
|96.57
|668.77
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|113,193,595
|85,920,795
|24.09%
|106.8
|70.96
|294.52
|style="text-align:left" | chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|85,920,795
|78,674,774
|8.43%
|28.9
|69.62
|825.51
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|78,674,774
|73,228,547
|6.92%
|68.7
|22.02
|318.04
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|73,228,547
|67,014,897
|8.49%
|23.2
|74.50
|878.02
|style="text-align:left" | chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|67,014,897
|57,625,231
|14.01%
|75.6
|34.49
|246.13
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|57,625,231
|48,070,606
|16.58%
|645.4
|4.11
|24.80
|style="text-align:left" | chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 30000 NG_n 10 run
|-
|style="text-align:left" |Terry Ligocki
|48,070,606
|44,254,286
|7.94%
|166.3
|6.38
|80.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 6 H 1 tH 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|44,254,286
|40,836,159
|7.72%
|188.3
|5.04
|65.29
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|40,836,159
|37,460,692
|8.27%
|192.3
|4.88
|58.99
|style="text-align:left" |
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 2 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 3 n 2 run 
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 3000 NG_n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 4 H 1 tH 1 n 2 run 
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 3 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 2 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|37,460,692
|36,167,570
|3.45%
|237.7
|1.51
|43.77
|style="text-align:left" |
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 3 H 0 tH 1 n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 10 chr_H 8 MitM_CTL NG maxT 10000 NG_n 4 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 2 run 
|-
|style="text-align:left" |Terry Ligocki
|36,167,570
|34,642,544
|4.22%
|467.2
|0.91
|21.50
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,642,544
|34,339,943
|0.87%
|383.1
|0.22
|25.12
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 0 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,339,943
|33,860,069
|1.40%
|666.5
|0.20
|14.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 2 tH 2 n 8 run
|-
|style="text-align:center" |'''Cumulative'''
|'''460,916,384'''
|'''33,860,069'''
|'''92.70%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

=== Stage 2 ===

Starting from the results of Stage 1, Terry Ligocki ran a variety of enumeration and decider codes. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~33.9M TMs to ~9.4M TMs (a 72.2% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|33,860,069
|21,065,769
|37.79%
|93.0
|38.20
|101.11
|style="text-align:left" |lr_enum_continue 4x3.in.txt 1000000 4x3.halt.txt 4x3.inf.txt 4x3.holdouts.txt 00 false
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|21,065,769
|18,949,009
|10.05%
|5,566.1
|0.11
|1.05
|style="text-align:left" |Enumerate.py max-loops 100_000 block-size 2 --tape-limit 1_000 --no-steps --time 1.0 --recursive --exp-linear-rules --lin-steps 0 --no-ctl --no-reverse-engineer --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,949,009
|18,138,027
|4.28%
|0.4
|511.59
|11953.46
|style="text-align:left" |Reverse_Engineer_Filter.py --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,138,027
|11,985,999
|33.92%
|4.8
|352.73
|1,039.95
|style="text-align:left" | chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|11,985,999
|9,988,715
|16.66%
|640.4
|0.87
|5.20
|style="text-align:left" |
chr_LRUH 24 chr_H 16 MitM_CTL NG maxT 30000 NG_n 3 run 
chr_LRUH 14 chr_H 2 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 5 run 
chr_asth 0 chr_LRUH 48 chr_H 48 MitM_CTL NG maxT 30000 NG_n 5 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 3 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 1 n 4 run 
chr_LRUH 14 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 0 n 6 run 
chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 18 chr_H 16 MitM_CTL NG maxT 30000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 1 tH 0 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 3 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|9,988,715
|9,401,447
|5.88%
|1,398.7
|0.12
|1.98
|style="text-align:left" |
chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 100000 NG_n 5 run 
chr_LRUH 22 chr_H 12 MitM_CTL NG maxT 100000 NG_n 6 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 100000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 16 H 1 tH 0 n 10 run 
chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 1000000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 6 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 2 tH 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 6 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 1 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 5 n 2 run 
|-
|style="text-align:center" |'''Cumulative'''
|'''33,860,069'''
|'''9,401,447'''
|'''72.23%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

==== Stage 3 ====
Starting from the results of Stage 2, Andrew Ducharme ran a variety of Ligocki and @mxdys deciders. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~9.4M TMs to ~5.6M (a 40.0% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:
{| 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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
| style="text-align:left" |Andrew Ducharme
|9,401,447
|7,753,702
|17.53%
|988.7
|
|
| style="text-align:left" |Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000 --block-mult=3 --time=0.5 --lin-steps=0 --no-ctl
| rowspan="28" style="text-align:left" |[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_?usp=drive_link Google Drive]
|-
| style="text-align:left" |Andrew Ducharme
|7,753,702
|7,409,705
|4.44%
|~500
|
|
| style="text-align:left" |lr_enum_continue 10000000 (10M steps)
|-
| style="text-align:left" |Andrew Ducharme
|7,409,705
|7,192,937
|2.93%
|1858.9
|
|
| style="text-align:left" |Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000 --block-mult=12 --time=1 --tape-limit=500 --lin-steps=0 --no-ctl
|-
| style="text-align:left" |Andrew Ducharme
|7,192,937
|6,711,936
|6.69%
|3.6
|
|
| style="text-align:left" | FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 2
FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 4
|-
| style="text-align:left" |Andrew Ducharme
|6,711,936
|6,506,888
|3.05%
|~500
|
|
| style="text-align:left" |
FAR CPS_LRU maxT 100000 LRUH [1,2] remaining parameters
|-
| style="text-align:left" |Andrew Ducharme
|6,506,888
|6,298,166
|3.21%
|~2200
|
|
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH [3,4]
|-
|Andrew Ducharme
|6,298,166
|6,257,722
|0.64%
|~2000
|
|
|FAR CPS_LRU maxT 100000 LRUH 5
|-
|Andrew Ducharme
|6,257,722
|6,237,675
|0.32%
|~2400
|
|
|FAR CPS_LRU maxT 100000 LRUH 6
|-
|Andrew Ducharme
|6,237,675
|6,156,619
|1.30%
|1798.3
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=250_000 --block-mult=1 --time=1 --tape-limit=1000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|6,156,619
|6,123,679
|0.54%
|1784.3
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=250_000 --block-mult=5 --time=1 --tape-limit=1000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|6,123,679
|6,071,297
|0.86%
|~7500
|
|
|FAR CPS_LRU maxT 100000 LRUH 12
|-
|Andrew Ducharme
|6,071,297
|5,913,070
|2.61%
|~25000
|
|
|FAR CPS_LRU maxT 1000000 LRUH [1,2]
|-
|Andrew Ducharme
|5,913,070
|5,718,346
|3.29%
|15790.2
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=1_000_000 --block-mult=8 --time=10 --tape-limit=5000 --max-steps-per-macro=1_000_000 --lin-steps=0 --no-ctl
|-
|Andrew Ducharme
|5,718,346
|5,641,006
|1.35%
|15989.4
|
|
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=10_000_000 --block-mult=3 --time=10 --tape-limit=5000 --max-steps-per-macro=1_000_000 --lin-steps=0 --no-ctl
|-
| style="text-align:center" |'''Cumulative'''
|'''9,401,447'''
|'''5,641,006'''
|'''40.00%'''
| ---
| ---
| ---
| style="text-align:left" | ---
|}

==References==

[[Category:BB Domains]][[Category:BB(4,3)]]

User:ADucharme

2026-04-25T22:04:00Z

ADucharme: /* BB(4,3) */

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts since that enumeration. I've also tried my hand at the analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last ~1500 BB(6) holdouts, I solved 67 and counting. Partial credit for some of these machines goes to Peacemaker II, who identifies permutations of machines I solved in the holdout list. Because of the shared behavior between permutations, I can apply the decider which solved to the original TM I found to the permutations, and often solve permutations too.

Solved halting TMs (49) with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs (18) with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9
1RB0RB_1LC1RA_0LA1RD_1LA1LE_1LF1LD_---0LC FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 3 n 12
1RB0LD_1RC0RE_0LA0RC_1LA1LD_0RF1RA_---1RC FAR CPS_LRU maxT 10000000 LRUH 6 H 3 tH 0 n 9
1RB1LB_1LC1RE_0RD0LB_0LB1RA_1LA0RF_---0RC FAR CPS_LRU maxT 10000000 LRUH 7 H 3 tH 1 n 4
1RB0LD_0RC1RF_1RD0RA_1LE1RB_1LC0LE_1RC--- FAR CPS_LRU maxT 10000000 LRUH 7 H 4 tH 1 n 24
1RB0LA_0RC---_1RD1RE_1LA1LD_1RD0RF_0RC1RC FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB0LA_1RC1RA_0LD1LA_1LF1RE_0RD0RE_0LC--- FAR RWL_mod maxT 10000000 H 4 mod 1 n 8
1RB1RF_1LC1LB_---0LD_1RE0LD_0RA1RA_0LE0RE FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB1RD_0RC1RE_1LD0RE_1LB---_0RA1LF_0LE0LF FAR CPS_LRU maxT 1000000 LRUH 32 H 1 tH 29 n 12
1RB1RE_1LC0RF_1RE0LD_1LC0LB_1RA0RE_1RC--- FAR CPS_LRU maxT 1000000 LRUH 32 H 4 tH 20 n 24
1RB0RE_1LC1RA_0LA1LD_1RE1LC_0RF1RB_---0LC FAR CPS_LRU maxT 1000000 LRUH 17 H 4 tH 13 n 3

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Games ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain. These remain my best-ever entries in these particular Busy Beaver games.

==== BB(7) ====
{| class="wikitable sortable"
|Place
|TM
|Score
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|5
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG}}
|10 ↑↑ 403.84
|-
|9
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}}
|10 ↑↑ 243.88
|}

{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}} was a bit of co-discovery: Iijil first enumerated the TM and I first showed it was halting.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}

==== BB(4,3) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|4
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|~10 ↑↑ 1023.47221
|-
|5
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 619.07737
|-
|6
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|~10 ↑↑ 512.10945
|-
|7
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|~10 ↑↑ 439.02781
|-
|8
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10 ↑↑ 234.06408
|}

User:ADucharme

2026-04-25T22:03:04Z

ADucharme: /* Busy Beaver Games */ add BB(4,3) top 10 TMs

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts since that enumeration. I've also tried my hand at the analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last ~1500 BB(6) holdouts, I solved 67 and counting. Partial credit for some of these machines goes to Peacemaker II, who identifies permutations of machines I solved in the holdout list. Because of the shared behavior between permutations, I can apply the decider which solved to the original TM I found to the permutations, and often solve permutations too.

Solved halting TMs (49) with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs (18) with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9
1RB0RB_1LC1RA_0LA1RD_1LA1LE_1LF1LD_---0LC FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 3 n 12
1RB0LD_1RC0RE_0LA0RC_1LA1LD_0RF1RA_---1RC FAR CPS_LRU maxT 10000000 LRUH 6 H 3 tH 0 n 9
1RB1LB_1LC1RE_0RD0LB_0LB1RA_1LA0RF_---0RC FAR CPS_LRU maxT 10000000 LRUH 7 H 3 tH 1 n 4
1RB0LD_0RC1RF_1RD0RA_1LE1RB_1LC0LE_1RC--- FAR CPS_LRU maxT 10000000 LRUH 7 H 4 tH 1 n 24
1RB0LA_0RC---_1RD1RE_1LA1LD_1RD0RF_0RC1RC FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB0LA_1RC1RA_0LD1LA_1LF1RE_0RD0RE_0LC--- FAR RWL_mod maxT 10000000 H 4 mod 1 n 8
1RB1RF_1LC1LB_---0LD_1RE0LD_0RA1RA_0LE0RE FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB1RD_0RC1RE_1LD0RE_1LB---_0RA1LF_0LE0LF FAR CPS_LRU maxT 1000000 LRUH 32 H 1 tH 29 n 12
1RB1RE_1LC0RF_1RE0LD_1LC0LB_1RA0RE_1RC--- FAR CPS_LRU maxT 1000000 LRUH 32 H 4 tH 20 n 24
1RB0RE_1LC1RA_0LA1LD_1RE1LC_0RF1RB_---0LC FAR CPS_LRU maxT 1000000 LRUH 17 H 4 tH 13 n 3

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Games ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain. These remain my best-ever entries in these particular Busy Beaver games.

==== BB(7) ====
{| class="wikitable sortable"
|Place
|TM
|Score
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|5
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG}}
|10 ↑↑ 403.84
|-
|9
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}}
|10 ↑↑ 243.88
|}

{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}} was a bit of co-discovery: Iijil first enumerated the TM and I first showed it was halting.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}

==== BB(4,3) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|4
|<code>1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD</code> (bbch)
|~10 ↑↑ 1023.47221
|-
|5
|<code>1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ</code> (bbch)
|~10 ↑↑ 619.07737
|-
|6
|<code>1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA</code> (bbch)
|~10 ↑↑ 512.10945
|-
|7
|<code>1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA</code> (bbch)
|~10 ↑↑ 439.02781
|-
|8
|<code>1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ</code> (bbch)
|~10 ↑↑ 234.06408
|}

BB(4,3)

2026-04-25T21:59:32Z

ADucharme: /* Top Halters */ templating TMs

The Busy Beaver problem for 4 states and 3 symbols is unsolved. The existence of [[Cryptids]] in the domain is given by the discovery of [[Bigfoot]] in [[BB(3,3)]]. The current [[Champions#3-Symbol TMs|champion]] is {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} which was discovered by Pavel Kropitz in May 2024 along with 6 other long running machines. It was [[User:Polygon/Page for analyses#1RB1RD1LC 2LB1RB1LC 1RZ1LA1LD 0RB2RA2RD (bbch)|analyzed by Polygon]] in Oct 2025, demonstrating the lower bounds:

<math display="block">S(4,3) > \Sigma(4,3) > 10 \uparrow^{4} 4</math>

== Top Halters ==
The longest running halting BB(4,3) TMs are split amongst two classes: the pentational and hexational TMs found by Pavel Kropitz outlined in the Potential Champions section, and the tetrational TMs found by comprehensive holdout filtering by Terry Ligocki. 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 longest running halters found by Pavel Kropitz are:
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|}
The top 20 scoring halting machines found by comprehensive search are:
{| class="wikitable"
|+
!Standard format
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD|halt}}
|~10↑↑1023.47221
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10↑↑619.07737
|Andrew Ducharme
|-
|{{TM|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA|halt}}
|~10↑↑512.10945
|Andrew Ducharme
|-
|{{TM|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA|halt}}
|~10↑↑439.02781
|Andrew Ducharme
|-
|{{TM|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ|halt}}
|~10↑↑234.06408
|Andrew Ducharme
|-
|{{TM|1RB0LC1RC_1LA2RB1LB_1RC2LA0RD_2LB1RZ2LC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LA1RA_1LA0RC1LC_1LC2RB0LD_2RA1RZ2RC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_2RC1LB2RD_1LD2LA0LB_0LA1RZ0LC|halt}}
|~10 ↑↑ 178.48320
|Andrew Ducharme
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA1LB_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB1LD0RC_2LC0RB1RA_1RA0LB1RD_0LA2LA1RZ|halt}}
|~10 ↑↑ 158.81916
|Andrew Ducharme
|-
|{{TM|1RB1RC1RB_1LC0RA2LD_2RA0LD1RZ_0LB2LD1RD|halt}}
|~10 ↑↑ 154.52968
|Andrew Ducharme
|-
|{{TM|1RB1LA1RD_2LA0LC2LD_1RZ2RA2LB_0LC2RC1RA|halt}}
|~10 ↑↑ 147.26175
|Andrew Ducharme
|-
|{{TM|1RB0RB1LC_2LC0LD1RA_2RB2LD1RZ_2LA2LB0LD|halt}}
|~10 ↑↑ 141.44248
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ2LD1RA|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|{{TM|1RB0RC1LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA---|halt}}
|~10 ↑↑ 139.06217
|Andrew Ducharme
|}

== Potential Champions ==
In May 2024, [https://discord.com/channels/960643023006490684/1026577255754903572/1243253180297646120 Pavel Kropitz found 7 halting TMs] that run for a large number of steps. Four of these are equivalent and were [https://discord.com/channels/960643023006490684/1331570843829932063/1337228898068463718 analyzed by Racheline] in February 2025, while the remaining three were [[User:Polygon/Page for analyses|analyzed by Polygon in October 2025.]]
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}*
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}
|<math>10 \uparrow\uparrow 9.873987</math>
|}
<nowiki>*</nowiki>equivalent to {{TM|0RB1RZ1RC_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}, {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ0RA|halt}} and {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ1RB|halt}}.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in December 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~633B(illion) TMs of which ~34.4B TMs were holdouts. Also found were ~206B halting TMs and ~392B infinite TMs. The number of holdouts was reduced to ~461M TMs (a 98.66% reduction).

Two C++ programs were run before the filters in the table.
<pre>
lr_enum 4 3 8 /dev/null /dev/null 4x3.unk.txt false
00 <= XX < 47: lr_enum_continue 4x3.in.XX 1000 /dev/null /dev/null 4x3.unk.txt.XX XX false
</pre>
Both do the initial enumeration and simple filtering. The "/dev/null" in both commands would be files where the halting and infinite TMs would be stored. The first command generates the TMs from a TNF tree for BB(4,3) of depth 8 and outputs the holdouts to 4x3.unk.txt. This file was then divided into 48 pieces, 4x3.in.XX, 0 <= XX < 47. The second commands (one for each XX) continues the enumeration by running each TM for 1,000 steps. It classifies each as halting, infinite, or unknown/holdout. Again, the halting and infinite TMs are "written" to /dev/null, i.e., they aren't saved. The holdouts are stored in 48 files: 4x3.unk.txt.XX.

For these runs the first command generated a total of ~45M TMs: ~1.86M halting, ~774K infinite, and ~42.0M holdouts. The second took the ~42.0M holdout TMs and generated a total of ~633B TMs: ~206B halting, ~392B infinite, and ~34.4B holdouts. These holdouts were used as a starting point of the filters below.

The "Description" column in the table below contain the command run. Two options are not given, "--infile=..." and an "--outfile=...". These are necessary and specify where to read and write the results, respectively. Note: The work flow was to divide the input holdouts into 48 pieces, run the command on each piece simultaneously on one of 48 cores, and then combine the 48 results into a group of holdouts.

The details are given in this table:

(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"
!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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|34,413,860,527
|30,874,934,791
|10.28%
|646.6
|1,520.36
|14,784.57
|style="text-align:left" |Reverse_Engineer_Filter.py
|rowspan="10" style="text-align:left" |[https://drive.google.com/drive/folders/1KMOVgngtUVMEA7EjxtNcsgksQ5Y4tby9?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|30,874,934,791
|12,942,386,396
|58.08%
|4,134.8
|1,204.72
|2,074.19
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|12,942,386,396
|4,534,322,415
|64.97%
|3,361.1
|694.88
|1,069.62
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|4,534,322,415
|2,959,598,830
|34.73%
|3,318.1
|131.83
|379.59
|style="text-align:left" |CPS_Filter.py --block-size=3
|-
|style="text-align:left" |Terry Ligocki
|2,959,598,830
|1,651,940,618
|44.18%
|2,700.6
|134.50
|304.42
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --no-steps --time=0.002 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|1,651,940,618
|854,984,279
|48.24%
|2,276.3
|97.25
|201.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.005 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|854,984,279
|683,163,325
|20.10%
|430.1
|110.96
|552.15
|style="text-align:left" |CPS_Filter.py --block-size=4 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|683,163,325
|460,916,384
|32.53%
|5,507.9
|11.21
|34.45
|style="text-align:left" |CPS_Filter.py --min-block-size=1 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:center" |'''Cumulative'''
|'''632,656,365,801'''
|'''460,916,384'''
|'''98.66%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

== Phase 2 ==

When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===

Starting from the results of Phase 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 33 variations were run. The holdouts were reduced from ~461B TMs to ~33.9M TMs (a 92.7% reduction). The details are given in the table below, including links to the Google Drive with the holdouts. Entries with multiple lines represent runs where all the commands in the "Description" were applied during one run.

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|460,916,384
|234,834,703
|49.05%
|96.7
|649.48
|1,324.10
|style="text-align:left" | chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 1000 NG_n 2 run
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1tFtg1eFC-AdqCzh7XNmx5O2mTQwtaNbm?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|234,834,703
|160,518,206
|31.65%
|70.9
|291.33
|920.57
|style="text-align:left" | chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 1000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|160,518,206
|132,296,033
|17.58%
|41.5
|188.86
|1,074.17
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 6 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|132,296,033
|113,193,595
|14.44%
|54.9
|96.57
|668.77
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|113,193,595
|85,920,795
|24.09%
|106.8
|70.96
|294.52
|style="text-align:left" | chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|85,920,795
|78,674,774
|8.43%
|28.9
|69.62
|825.51
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|78,674,774
|73,228,547
|6.92%
|68.7
|22.02
|318.04
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|73,228,547
|67,014,897
|8.49%
|23.2
|74.50
|878.02
|style="text-align:left" | chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|67,014,897
|57,625,231
|14.01%
|75.6
|34.49
|246.13
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|57,625,231
|48,070,606
|16.58%
|645.4
|4.11
|24.80
|style="text-align:left" | chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 30000 NG_n 10 run
|-
|style="text-align:left" |Terry Ligocki
|48,070,606
|44,254,286
|7.94%
|166.3
|6.38
|80.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 6 H 1 tH 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|44,254,286
|40,836,159
|7.72%
|188.3
|5.04
|65.29
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|40,836,159
|37,460,692
|8.27%
|192.3
|4.88
|58.99
|style="text-align:left" |
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 2 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 3 n 2 run 
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 3000 NG_n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 4 H 1 tH 1 n 2 run 
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 3 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 2 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|37,460,692
|36,167,570
|3.45%
|237.7
|1.51
|43.77
|style="text-align:left" |
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 3 H 0 tH 1 n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 10 chr_H 8 MitM_CTL NG maxT 10000 NG_n 4 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 2 run 
|-
|style="text-align:left" |Terry Ligocki
|36,167,570
|34,642,544
|4.22%
|467.2
|0.91
|21.50
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,642,544
|34,339,943
|0.87%
|383.1
|0.22
|25.12
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 0 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,339,943
|33,860,069
|1.40%
|666.5
|0.20
|14.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 2 tH 2 n 8 run
|-
|style="text-align:center" |'''Cumulative'''
|'''460,916,384'''
|'''33,860,069'''
|'''92.70%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

=== Stage 2 ===

Starting from the results of Phase 2 Stage, Terry Ligocki ran a variety of enumeration and decider codes. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~33.9M TMs to ~9.4M TMs (a 72.2% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|33,860,069
|21,065,769
|37.79%
|93.0
|38.20
|101.11
|style="text-align:left" |lr_enum_continue 4x3.in.txt 1000000 4x3.halt.txt 4x3.inf.txt 4x3.holdouts.txt 00 false
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|21,065,769
|18,949,009
|10.05%
|5,566.1
|0.11
|1.05
|style="text-align:left" |Enumerate.py max-loops 100_000 block-size 2 --tape-limit 1_000 --no-steps --time 1.0 --recursive --exp-linear-rules --lin-steps 0 --no-ctl --no-reverse-engineer --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,949,009
|18,138,027
|4,28%
|0.4
|511.59
|11953.46
|style="text-align:left" |Reverse_Engineer_Filter.py --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,138,027
|11,985,999
|33.92%
|4.8
|352.73
|1,039.95
|style="text-align:left" | chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|11,985,999
|9,988,715
|16.66%
|640.4
|0.87
|5.20
|style="text-align:left" |
chr_LRUH 24 chr_H 16 MitM_CTL NG maxT 30000 NG_n 3 run 
chr_LRUH 14 chr_H 2 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 5 run 
chr_asth 0 chr_LRUH 48 chr_H 48 MitM_CTL NG maxT 30000 NG_n 5 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 3 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 1 n 4 run 
chr_LRUH 14 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 0 n 6 run 
chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 18 chr_H 16 MitM_CTL NG maxT 30000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 1 tH 0 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 3 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|9,988,715
|9,401,447
|5.88%
|1,398.7
|0.12
|1.98
|style="text-align:left" |
chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 100000 NG_n 5 run 
chr_LRUH 22 chr_H 12 MitM_CTL NG maxT 100000 NG_n 6 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 100000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 16 H 1 tH 0 n 10 run 
chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 1000000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 6 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 2 tH 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 6 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 1 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 5 n 2 run 
|-
|style="text-align:center" |'''Cumulative'''
|'''33,860,069'''
|'''9,401,447'''
|'''72.23%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

==References==

[[Category:BB Domains]][[Category:BB(4,3)]]

BB(4,3)

2026-04-25T21:57:58Z

ADucharme: /* Top Halters */ adding a bunch of new top 20 halters found by comprehensive search

The Busy Beaver problem for 4 states and 3 symbols is unsolved. The existence of [[Cryptids]] in the domain is given by the discovery of [[Bigfoot]] in [[BB(3,3)]]. The current [[Champions#3-Symbol TMs|champion]] is {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} which was discovered by Pavel Kropitz in May 2024 along with 6 other long running machines. It was [[User:Polygon/Page for analyses#1RB1RD1LC 2LB1RB1LC 1RZ1LA1LD 0RB2RA2RD (bbch)|analyzed by Polygon]] in Oct 2025, demonstrating the lower bounds:

<math display="block">S(4,3) > \Sigma(4,3) > 10 \uparrow^{4} 4</math>

== Top Halters ==
The longest running halting BB(4,3) TMs are split amongst two classes: the pentational and hexational TMs found by Pavel Kropitz outlined in the Potential Champions section, and the tetrational TMs found by comprehensive holdout filtering by Terry Ligocki. 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 longest running halters found by Pavel Kropitz are:
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|}
The top 20 scoring halting machines found by comprehensive search are:
{| class="wikitable"
|+
!Standard format
!Approximate sigma score
!Discoverer
|-
|1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD
|~10↑↑1023.47221
|Andrew Ducharme
|-
|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ
|~10↑↑619.07737
|Andrew Ducharme
|-
|1RB1RZ2RD_1LC0RD0RC_2LC1LA0RB_2RC0RC2RA
|~10↑↑512.10945
|Andrew Ducharme
|-
|1RB1RZ0RC_1RC1RA0LD_2RD2RB0RD_1LB2LD2RA
|~10↑↑439.02781
|Andrew Ducharme
|-
|1RB0LC1RD_1RC1LD0RA_2LA0RC1RB_0LB2LB1RZ
|~10↑↑234.06408
|Andrew Ducharme
|-
|{{TM|1RB0LC1RC_1LA2RB1LB_1RC2LA0RD_2LB1RZ2LC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LA1RA_1LA0RC1LC_1LC2RB0LD_2RA1RZ2RC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|1RB2LC1RA_2RC1LB2RD_1LD2LA0LB_0LA1RZ0LC
|~10 ↑↑ 178.48320
|Andrew Ducharme
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA1LB_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|1RB1LD0RC_2LC0RB1RA_1RA0LB1RD_0LA2LA1RZ
|~10 ↑↑ 158.81916
|Andrew Ducharme
|-
|1RB1RC1RB_1LC0RA2LD_2RA0LD1RZ_0LB2LD1RD
|~10 ↑↑ 154.52968
|Andrew Ducharme
|-
|1RB1LA1RD_2LA0LC2LD_1RZ2RA2LB_0LC2RC1RA
|~10 ↑↑ 147.26175
|Andrew Ducharme
|-
|{{TM|1RB0RB1LC_2LC0LD1RA_2RB2LD1RZ_2LA2LB0LD}}
|~10 ↑↑ 141.44248
|Andrew Ducharme
|-
|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA1RA
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|1RB0RC2LB_2LC2RD1LC_1RC0LC1LB_1RZ2LD1RA
|~10 ↑↑ 139.06217
|Andrew Ducharme
|-
|1RB0RC1LB_2LC2RD1LC_1RC0LC1LB_1RZ1RA---
|~10 ↑↑ 139.06217
|Andrew Ducharme
|}

== Potential Champions ==
In May 2024, [https://discord.com/channels/960643023006490684/1026577255754903572/1243253180297646120 Pavel Kropitz found 7 halting TMs] that run for a large number of steps. Four of these are equivalent and were [https://discord.com/channels/960643023006490684/1331570843829932063/1337228898068463718 analyzed by Racheline] in February 2025, while the remaining three were [[User:Polygon/Page for analyses|analyzed by Polygon in October 2025.]]
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}*
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}
|<math>10 \uparrow\uparrow 9.873987</math>
|}
<nowiki>*</nowiki>equivalent to {{TM|0RB1RZ1RC_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}, {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ0RA|halt}} and {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ1RB|halt}}.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in December 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~633B(illion) TMs of which ~34.4B TMs were holdouts. Also found were ~206B halting TMs and ~392B infinite TMs. The number of holdouts was reduced to ~461M TMs (a 98.66% reduction).

Two C++ programs were run before the filters in the table.
<pre>
lr_enum 4 3 8 /dev/null /dev/null 4x3.unk.txt false
00 <= XX < 47: lr_enum_continue 4x3.in.XX 1000 /dev/null /dev/null 4x3.unk.txt.XX XX false
</pre>
Both do the initial enumeration and simple filtering. The "/dev/null" in both commands would be files where the halting and infinite TMs would be stored. The first command generates the TMs from a TNF tree for BB(4,3) of depth 8 and outputs the holdouts to 4x3.unk.txt. This file was then divided into 48 pieces, 4x3.in.XX, 0 <= XX < 47. The second commands (one for each XX) continues the enumeration by running each TM for 1,000 steps. It classifies each as halting, infinite, or unknown/holdout. Again, the halting and infinite TMs are "written" to /dev/null, i.e., they aren't saved. The holdouts are stored in 48 files: 4x3.unk.txt.XX.

For these runs the first command generated a total of ~45M TMs: ~1.86M halting, ~774K infinite, and ~42.0M holdouts. The second took the ~42.0M holdout TMs and generated a total of ~633B TMs: ~206B halting, ~392B infinite, and ~34.4B holdouts. These holdouts were used as a starting point of the filters below.

The "Description" column in the table below contain the command run. Two options are not given, "--infile=..." and an "--outfile=...". These are necessary and specify where to read and write the results, respectively. Note: The work flow was to divide the input holdouts into 48 pieces, run the command on each piece simultaneously on one of 48 cores, and then combine the 48 results into a group of holdouts.

The details are given in this table:

(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"
!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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|34,413,860,527
|30,874,934,791
|10.28%
|646.6
|1,520.36
|14,784.57
|style="text-align:left" |Reverse_Engineer_Filter.py
|rowspan="10" style="text-align:left" |[https://drive.google.com/drive/folders/1KMOVgngtUVMEA7EjxtNcsgksQ5Y4tby9?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|30,874,934,791
|12,942,386,396
|58.08%
|4,134.8
|1,204.72
|2,074.19
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|12,942,386,396
|4,534,322,415
|64.97%
|3,361.1
|694.88
|1,069.62
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|4,534,322,415
|2,959,598,830
|34.73%
|3,318.1
|131.83
|379.59
|style="text-align:left" |CPS_Filter.py --block-size=3
|-
|style="text-align:left" |Terry Ligocki
|2,959,598,830
|1,651,940,618
|44.18%
|2,700.6
|134.50
|304.42
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --no-steps --time=0.002 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|1,651,940,618
|854,984,279
|48.24%
|2,276.3
|97.25
|201.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.005 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|854,984,279
|683,163,325
|20.10%
|430.1
|110.96
|552.15
|style="text-align:left" |CPS_Filter.py --block-size=4 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|683,163,325
|460,916,384
|32.53%
|5,507.9
|11.21
|34.45
|style="text-align:left" |CPS_Filter.py --min-block-size=1 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:center" |'''Cumulative'''
|'''632,656,365,801'''
|'''460,916,384'''
|'''98.66%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

== Phase 2 ==

When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===

Starting from the results of Phase 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 33 variations were run. The holdouts were reduced from ~461B TMs to ~33.9M TMs (a 92.7% reduction). The details are given in the table below, including links to the Google Drive with the holdouts. Entries with multiple lines represent runs where all the commands in the "Description" were applied during one run.

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|460,916,384
|234,834,703
|49.05%
|96.7
|649.48
|1,324.10
|style="text-align:left" | chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 1000 NG_n 2 run
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1tFtg1eFC-AdqCzh7XNmx5O2mTQwtaNbm?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|234,834,703
|160,518,206
|31.65%
|70.9
|291.33
|920.57
|style="text-align:left" | chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 1000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|160,518,206
|132,296,033
|17.58%
|41.5
|188.86
|1,074.17
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 6 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|132,296,033
|113,193,595
|14.44%
|54.9
|96.57
|668.77
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|113,193,595
|85,920,795
|24.09%
|106.8
|70.96
|294.52
|style="text-align:left" | chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|85,920,795
|78,674,774
|8.43%
|28.9
|69.62
|825.51
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|78,674,774
|73,228,547
|6.92%
|68.7
|22.02
|318.04
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|73,228,547
|67,014,897
|8.49%
|23.2
|74.50
|878.02
|style="text-align:left" | chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|67,014,897
|57,625,231
|14.01%
|75.6
|34.49
|246.13
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|57,625,231
|48,070,606
|16.58%
|645.4
|4.11
|24.80
|style="text-align:left" | chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 30000 NG_n 10 run
|-
|style="text-align:left" |Terry Ligocki
|48,070,606
|44,254,286
|7.94%
|166.3
|6.38
|80.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 6 H 1 tH 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|44,254,286
|40,836,159
|7.72%
|188.3
|5.04
|65.29
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|40,836,159
|37,460,692
|8.27%
|192.3
|4.88
|58.99
|style="text-align:left" |
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 2 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 3 n 2 run 
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 3000 NG_n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 4 H 1 tH 1 n 2 run 
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 3 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 2 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|37,460,692
|36,167,570
|3.45%
|237.7
|1.51
|43.77
|style="text-align:left" |
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 3 H 0 tH 1 n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 10 chr_H 8 MitM_CTL NG maxT 10000 NG_n 4 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 2 run 
|-
|style="text-align:left" |Terry Ligocki
|36,167,570
|34,642,544
|4.22%
|467.2
|0.91
|21.50
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,642,544
|34,339,943
|0.87%
|383.1
|0.22
|25.12
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 0 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,339,943
|33,860,069
|1.40%
|666.5
|0.20
|14.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 2 tH 2 n 8 run
|-
|style="text-align:center" |'''Cumulative'''
|'''460,916,384'''
|'''33,860,069'''
|'''92.70%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

=== Stage 2 ===

Starting from the results of Phase 2 Stage, Terry Ligocki ran a variety of enumeration and decider codes. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~33.9M TMs to ~9.4M TMs (a 72.2% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|33,860,069
|21,065,769
|37.79%
|93.0
|38.20
|101.11
|style="text-align:left" |lr_enum_continue 4x3.in.txt 1000000 4x3.halt.txt 4x3.inf.txt 4x3.holdouts.txt 00 false
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|21,065,769
|18,949,009
|10.05%
|5,566.1
|0.11
|1.05
|style="text-align:left" |Enumerate.py max-loops 100_000 block-size 2 --tape-limit 1_000 --no-steps --time 1.0 --recursive --exp-linear-rules --lin-steps 0 --no-ctl --no-reverse-engineer --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,949,009
|18,138,027
|4,28%
|0.4
|511.59
|11953.46
|style="text-align:left" |Reverse_Engineer_Filter.py --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,138,027
|11,985,999
|33.92%
|4.8
|352.73
|1,039.95
|style="text-align:left" | chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|11,985,999
|9,988,715
|16.66%
|640.4
|0.87
|5.20
|style="text-align:left" |
chr_LRUH 24 chr_H 16 MitM_CTL NG maxT 30000 NG_n 3 run 
chr_LRUH 14 chr_H 2 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 5 run 
chr_asth 0 chr_LRUH 48 chr_H 48 MitM_CTL NG maxT 30000 NG_n 5 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 3 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 1 n 4 run 
chr_LRUH 14 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 0 n 6 run 
chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 18 chr_H 16 MitM_CTL NG maxT 30000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 1 tH 0 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 3 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|9,988,715
|9,401,447
|5.88%
|1,398.7
|0.12
|1.98
|style="text-align:left" |
chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 100000 NG_n 5 run 
chr_LRUH 22 chr_H 12 MitM_CTL NG maxT 100000 NG_n 6 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 100000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 16 H 1 tH 0 n 10 run 
chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 1000000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 6 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 2 tH 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 6 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 1 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 5 n 2 run 
|-
|style="text-align:center" |'''Cumulative'''
|'''33,860,069'''
|'''9,401,447'''
|'''72.23%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

==References==

[[Category:BB Domains]][[Category:BB(4,3)]]

BB(7)

2026-04-06T20:20:20Z

ADucharme: /* Phase 2 */ 7x2 progress completely updated

The 7-state, 2-symbol Busy Beaver problem, '''BB(7)''', refers to the unsolved 7th 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 
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] 
[https://drive.google.com/drive/folders/1kJ6tlmX8_7AQ8qpR1mSQ-Fz4-fPfwYBn?usp=drive_link Terry Ligocki]
|-
|05-09xxxx
|
@Iijil 
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] 
[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 
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] 
[https://drive.google.com/drive/folders/1kJ6tlmX8_7AQ8qpR1mSQ-Fz4-fPfwYBn?usp=drive_link Terry Ligocki]
|-
|19xxxx
|
Katelyn Doucette 
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 
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] 
[https://drive.google.com/drive/folders/1lyYN2wznnrfM0dg-dKprHODeYaTxdtzP?usp=drive_link Terry Ligocki]
|-
|49xxxx
|
Tobiáš Brichta 
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] 
[https://drive.google.com/drive/folders/1kJ6tlmX8_7AQ8qpR1mSQ-Fz4-fPfwYBn?usp=drive_link Terry Ligocki]
|-
|50xxxx
|
@prurq 
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 
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] 
[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 
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] 
[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 
[https://drive.google.com/file/d/1LXUKxqRhwW_Q1QGxWuRheV3sZu18hgOK/view?usp=drive_link Holdouts] 
[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 
[https://drive.google.com/file/d/18G2ofUaMZIKFwNCNVHxTRjasV6p39Wr2/view?usp=drive_link Holdouts] 
[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 
[https://drive.google.com/file/d/1VEC5hum9Z9nkDhOhAw1bWsS8yLalrGWC/view?usp=drive_link Holdouts] 
[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 
[https://drive.google.com/file/d/1N-vMMupgrsmIXe38sFczZRcF0t8XvgLT/view?usp=drive_link Holdouts] 
[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 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 6 mod 2 n 12 run 
chr_LRUH 18 chr_H 16 MitM_CTL NG maxT 100000 NG_n 3 run 
MitM_CTL CPS_LRU sim 1001 maxT 1000000 LRUH 20 H 2 tH 4 n 2 run 
chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 1000000 NG_n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 4 n 4 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 9 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 12 mod 3 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 300000 H 3 mod 2 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 1 tH 1 n 4 run 
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 100000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 4 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 1 n 3 run 
|
|rowspan="1" style="text-align:left"|Stage 5 
[https://drive.google.com/file/d/1jTyRvblSJnRDTwsfSCqD7kK0T4zxgU2A/view?usp=drive_link Holdouts] 
[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="10" |[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
|18,306,497
|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
|18,306,497
|18,254,547
|0.28%
|~3200
|
|
|LRUH 1 H [0,1] tH [0,1] n [1-31]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1470510369687601375 discord]
|-
|Andrew Ducharme
|18,254,547
|18,195,716
|0.32%
|~4000
|
|
|maxT 100000 remaining parameters w/ LRUH 2
|[https://discord.com/channels/960643023006490684/1369339127652159509/1472115313683202189 discord]
|-
|Andrew Ducharme
|18,195,716
|18,037,417
|0.87%
|~2200
|
|
|maxT 100000 remaining parameters w/ LRUH 3
|[https://discord.com/channels/960643023006490684/1369339127652159509/1481889113232904295 discord]
|-
|Andrew Ducharme
|18,037,417
|17,823,260
|1.19%
|~7200
|
|
|maxT 100000 LRUH 4
|[https://discord.com/channels/960643023006490684/1369339127652159509/1489495196180942881 discord]
|-
| style="text-align:center" |'''Stage 6 Cumulative'''
|'''20,405,295'''
|'''17,823,260'''
|'''12.65%'''
|'''45809.7'''
| ---
| ---
| ---
| ---
|
|-
| style="text-align:center" |'''Overall Cumulative'''
|'''86,129,304'''
|'''17,823,260'''
|'''79.31%'''
|'''93019.8'''
| ---
| ---
| ---
| ---
|}

==References==
[[Category:BB Domains]][[Category:BB(7)]]
<references />

BB(2,6)

2026-04-05T15:50:22Z

ADucharme: update 2x6 reduction table, google drive

The 2-state, 6-symbol Busy Beaver problem, '''BB(2,6),''' is unsolved. With cryptids like [[Hydra]] in the preceding domain [[BB(2,5)]], we know that we must solve a [[Collatz-like]] problem in order to solve BB(2,6).

The current BB(2,6) champion {{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}} was discovered by Pavel Kropitz in May 2023, proving the lower bound:<math display="block">S(2,6) > \Sigma(2,6) > 10 \uparrow \uparrow 10 \uparrow\uparrow 10^{10^{115}} > 10 \uparrow \uparrow \uparrow 3</math>

== 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 20 highest known scoring machines are:
{| class="wikitable"
|+
!TM
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|10 ↑↑↑ 3
|Pavel Kropitz
|-
|{{TM|1RB2LA1RZ1RB5RB0RB_2LA4RA3LB5LB5RA4LB|halt}}
|10 ↑↑ 19892.08
|Peacemaker II
|-
|{{TM|1RB3LA4LB0RB1RA3LA_2LA2RA4LA1RA5RB1RZ|halt}}
|10 ↑↑ 91.17
|Pavel Kropitz
|-
|{{TM|1RB2LA1RA4LA5RA0LB_1LA3RA2RB1RZ3RB4LA|halt}}
|10 ↑↑ 70.27
|Shawn Ligocki
|-
|{{TM|1RB2LB1RZ3LA2LA4RB_1LA3RB4RB1LB5LB0RA|halt}}
|10 ↑↑ 69.68
|Shawn Ligocki
|-
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|Andrew Ducharme
|-
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|Andrew Ducharme
|-
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|Andrew Ducharme
|-
|{{TM|1RB3LB3RB4LA2LA4LA_2LA2RB1LB0RA5RA1RZ|halt}}
|10 ↑↑ 21.54
|Shawn Ligocki
|-
|{{TM|1RB2LB3LA1RA0RA1RZ_1LA2RB1LB4RB5RA3LA|halt}}
|10 ↑↑ 20.58
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB1RA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB5LA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Andrew Ducharme
|-
|{{TM|1RB3RA4LB5RA5LB4RA_2LA1RZ1RB2LA5LA0LA|halt}}
|10 ↑↑ 17.08
|Andrew Ducharme
|-
|{{TM|1RB3RA4LA1LA0LA1RZ_2LA0LB1RA1LB5LB2RA|halt}}
|10 ↑↑ 15.44
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ2LA|halt}}
|10 ↑↑ 14.35
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ3RA|halt}}
|10 ↑↑ 14.17
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ1LA|halt}}
|10 ↑↑ 14.05
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ0RA|halt}}
|10 ↑↑ 13.69
|Andrew Ducharme
|-
|{{TM|1RB3LA3RA4LB2LB0LA_2LA5LB2RB0RA0RA1RZ|halt}}
|10 ↑↑ 12.42
|Andrew Ducharme
|-
|{{TM|1RB0LB4LA2RA2RB1LB_2LA4LA3LB5LA1RA1RZ|halt}}
|10 ↑↑ 11.70
|Andrew Ducharme
|}
All decimal places are truncated.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in November 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~24B(illion) TMs of which ~2.278B were holdout TMs. This was reduced to ~22M holdout TMs (a 99.02% reduction). The details are given in this table, including links to the Google Drive with the holdouts and details of the computation:

(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"
!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" |Data
|-
|style="text-align:left" |Terry Ligocki
|2,278,655,696
|2,109,114,609
|7.44%
|40.9
|1,150.90
|15,468.23
|style="text-align:left" |Reverse_Engineer_Filter.py
|style="text-align:left", rowspan="100" |[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|2,109,114,609
|683,067,538
|67.61%
|452.8
|874.77
|1,293.79
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|683,067,538
|210,993,434
|69.11%
|396.4
|330.85
|478.72
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|210,993,434
|141,680,232
|32.85%
|273.9
|70.29
|213.97
|style="text-align:left" |CPS_Filter.py --block-size=3 --max_steps=10_000
|-
|style="text-align:left" |Terry Ligocki
|141,680,232
|66,029,536
|53.40%
|486.6
|43.18
|80.87
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --time=10 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|66,029,536
|46,119,004
|30.15%
|167.4
|33.05
|109.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.01 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|46,119,004
|39,034,142
|15.36%
|170.1
|11.57
|75.34
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=12 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|39,034,142
|29,109,512
|25.43%
|2,221.6
|1.24
|4.88
|style="text-align:left" |CPS_Filter.py --min-block-size=4 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:left" |Terry Ligocki
|29,109,512
|24,536,819
|15.71%
|384.2
|3.31
|21.05
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=6 --recursive --no-steps --time=0.05 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|24,536,819
|22,302,296
|9.11%
|1,047.5
|0.59
|6.51
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=4 --recursive --no-steps --time=1.00 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|}

== Phase 2 ==
When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===
Andrew Ducharme ran another pass of "lr_enum_continue" with the maximum number of steps set to 10 million. The holdouts were reduced from ~22.3M TMs to ~20.4M TMs (a 8.72% reduction). The entry in the table below has a rather technical/arcane/cryptic description. This was an effort to capture enough information to rerun that filter in parallel with specific C++ code, lr_enum_continue, and a specific parallel queuing system, Slurm:

(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"
!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" |Data
|-
|style="text-align:left" |Andrew Ducharme
|22,302,296
|20,358,011
|8.72%
|1,350.0
|0.40
|4.59
|style="text-align:left" |lr_enum_continue ${WORK_DIR}chunk_${SLURM_ARRAY_TASK_ID} 10000000 ${WORK_DIR}halt_${SLURM_ARRAY_TASK_ID}.txt ${WORK_DIR}inf_${SLURM_ARRAY_TASK_ID}.txt ${WORK_DIR}unknown_${SLURM_ARRAY_TASK_ID}.txt "" false
|style="text-align:left" rowspan="50"|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|}

=== Stage 2 ===
Starting from the results of Stage 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 50 variations were run. The holdouts were reduced from ~20.4M TMs to ~907K TMs (a 95.5% reduction). The details are given in this table, including links to the Google Drive with the holdouts and details of the computation:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|20,358,011
|19,500,847
|4.21%
|22.0
|10.84
|257.42
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 6 run
|style="text-align:left" rowspan="50"|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|19,500,847
|18,747,861
|3.86%
|86.0
|2.43
|63.01
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|18,747,861
|4,811,076
|74.34%
|47.0
|82.33
|110.75
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|4,811,076
|2,982,075
|38.02%
|17.1
|29.74
|78.22
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|2,982,075
|2,897,340
|2.84%
|15.2
|1.55
|54.64
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 8 mod 3 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|2,897,340
|2,850,781
|1.61%
|16.7
|0.77
|48.17
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n 7 run
|-
|style="text-align:left" |Terry Ligocki
|2,850,781
|2,759,635
|3.20%
|13.7
|1.85
|58.01
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|2,759,635
|1,953,426
|29.21%
|13.6
|16.48
|56.42
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,953,426
|1,855,545
|5.01%
|2.4
|11.18
|223.14
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,855,545
|1,647,269
|11.22%
|6.6
|8.80
|78.40
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,647,269
|1,608,166
|2.37%
|3.4
|3.20
|134.96
|style="text-align:left" |chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,608,166
|1,585,745
|1.39%
|9.6
|0.65
|46.35
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|1,585,745
|1,555,673
|1.90%
|5.7
|1.47
|77.73
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 10000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,555,673
|1,428,534
|8.17%
|9.3
|3.78
|46.31
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 2 tH 0 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,428,534
|1,340,964
|6.13%
|0.8
|29.70
|484.55
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 10000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,340,964
|1,286,439
|4.07%
|0.8
|18.40
|452.56
|style="text-align:left" |chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,286,439
|1,273,911
|0.97%
|0.8
|4.20
|430.88
|style="text-align:left" |chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,273,911
|1,265,198
|0.68%
|0.8
|2.88
|420.73
|style="text-align:left" |chr_LRUH 3 chr_H 1 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,265,198
|1,258,925
|0.50%
|0.9
|1.99
|400.83
|style="text-align:left" |chr_LRUH 8 chr_H 6 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,258,925
|1,242,136
|1.33%
|0.8
|5.51
|412.84
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 0 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,242,136
|1,231,731
|0.84%
|1.0
|2.78
|331.77
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 2 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,231,731
|1,216,646
|1.22%
|1.0
|4.15
|338.72
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 12 H 0 tH 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,216,646
|1,214,294
|0.19%
|0.9
|0.76
|393.03
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 2 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,214,294
|1,213,431
|0.07%
|0.9
|0.28
|391.30
|style="text-align:left" |chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,213,431
|1,211,390
|0.17%
|1.1
|0.52
|307.13
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 1 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,211,390
|1,209,989
|0.12%
|1.1
|0.35
|306.09
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 100000 NG_n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,209,989
|1,209,974
|0.00%
|0.9
|0.00
|381.42
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 16 H 1 tH 0 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,209,974
|1,201,890
|0.67%
|2.5
|0.90
|134.19
|style="text-align:left" |chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,201,890
|1,200,086
|0.15%
|1.3
|0.37
|248.36
|style="text-align:left" |chr_LRUH 10 chr_H 6 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,200,086
|1,199,734
|0.03%
|1.2
|0.08
|270.32
|style="text-align:left" |chr_asth 0 chr_LRUH 3 chr_H 3 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,199,734
|1,198,893
|0.07%
|2.3
|0.10
|147.66
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 2 mod 6 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,198,893
|1,165,493
|2.79%
|4.5
|2.05
|73.44
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 4 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|1,165,493
|1,153,863
|1.00%
|9.3
|0.35
|34.88
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 0 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,153,863
|1,144,711
|0.79%
|3.7
|0.69
|87.51
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 5 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,144,711
|1,127,789
|1.48%
|7.9
|0.60
|40.26
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 30000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,127,789
|1,124,762
|0.27%
|4.7
|0.18
|66.75
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 0 tH 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|1,124,762
|1,117,226
|0.67%
|5.6
|0.37
|55.36
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 12 H 0 tH 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,117,226
|1,109,057
|0.73%
|7.7
|0.30
|40.49
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,109,057
|1,083,097
|2.34%
|11.4
|0.63
|27.06
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 30000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,083,097
|1,077,833
|0.49%
|11.2
|0.13
|26.81
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 100000 NG_n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,077,833
|1,066,795
|1.02%
|24.1
|0.13
|12.40
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 2 tH 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,066,795
|1,039,229
|2.58%
|52.6
|0.15
|5.64
|style="text-align:left" |chr_LRUH 14 chr_H 6 MitM_CTL NG maxT 100000 NG_n 11 run
|-
|style="text-align:left" |Terry Ligocki
|1,039,229
|1,019,286
|1.92%
|43.5
|0.13
|6.63
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 12 mod 1 n 3 run
|-
|style="text-align:left" |Terry Ligocki
|1,019,286
|993,556
|2.52%
|66.8
|0.11
|4.24
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 8 H 2 tH 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|993,556
|985,718
|0.79%
|78.3
|0.03
|3.53
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 1 tH 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|985,718
|981,095
|0.47%
|83.7
|0.02
|3.27
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 6 H 1 tH 0 n 9 run
|-
|style="text-align:left" |Terry Ligocki
|981,095
|975,912
|0.53%
|79.4
|0.02
|3.43
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 16 mod 1 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|975,912
|974,180
|0.18%
|84.6
|0.01
|3.20
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 16 mod 4 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|974,180
|971,254
|0.30%
|96.9
|0.01
|2.79
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 100000 H 12 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|971,254
|970,101
|0.12%
|105.6
|0.00
|2.56
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 0 tH 0 n 18 run
|}

=== Stage 3 ===
Starting from the results of Stage 2, Andrew Ducharme ran "lr_enum_continue" with the maximum number of steps set to 100 million, then "Enumerate.py" with various parameters. A total of 10 Enumerate variations were run. The holdouts were reduced from ~970K TMs to ~867K TMs (a 10.63% reduction). The details are given in this table, including links to the Google Drive with the holdouts and details of the computation:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Andrew Ducharme
|970,101
|939,447
|3.16%
| --
| --
| --
|style="text-align:left" |lr_enum_continue 100_000_000 steps
| rowspan="11" |[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=drive_link Google Drive]
|-
| style="text-align:left" |Andrew Ducharme
|939,447
|903,224
|3.86%
|440.3
|0.03
|0.59
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=4 --no-ctl --lin-steps=0 --time=2 --force --save-freq=1000
|-
| style="text-align:left" |Andrew Ducharme
|903,224
|895,813
|0.82%
|647.7
|0.00
|0.39
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=3 --no-ctl --lin-steps=0 --time=3 --force --save-freq=1000
|-
| style="text-align:left" |Andrew Ducharme
|895,813
|889,838
|0.67%
|609.3
|0.00
|0.41
| style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=8 --no-ctl --lin-steps=0 --time=4 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|889,838
|880,278
|1.07%
|1,638.9
|0.00
|0.15
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=12 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|880,278
|877,485
|0.32%
|1,885.5
|0.00
|0.13
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=6 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|877,485
|875,062
|0.28%
|2,068.8
|0.00
|0.12
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=5 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|875,062
|873,469
|0.18%
|1,785.4
|0.00
|0.14
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=7 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|873,469
|870,085
|0.39%
|9,270.0
|0.00
|0.03
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=1_000_000 --block-mult=2 --tape-limit=500 --time=120 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|870,085
|869,001
|0.12%
|4,498.3
|0.00
|0.05
|style="text-align:left" |Enumerate.py --no-steps --exp-linear-rules --max_loops=10_000_000 --block-mult=60 --tape-limit=5000 --no-ctl --lin-steps=0 --force --save-freq=1000
|-
|style="text-align:left" |Andrew Ducharme
|869,001
|867,008
|0.23%
|3997.4
|0.00
|0.06
|style="text-align:left"|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=9 --tape-limit=5000 --max-steps-per-macro=100_000 --lin-steps=0 --no-ctl --force --save-freq=250
|}
The total time spent on the lr_enum_continue computation was not recorded.

=== Stage 4 ===
Following the release of @mxdys's implementation of FAR deciders in C++, these deciders were applied to the 2x6 holdouts by Andrew Ducharme. The details are given in this table, including links to the Google Drive with the holdouts and solved TMs per decider:

(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"
! colspan="2" |Holdout TMs
! rowspan="2" |<math>*^1</math>
! rowspan="2" |<math>*^2</math>
! colspan="2" |TMs/sec/core
! rowspan="2" |Description
! rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|867,008
|811,301
|6.43%
|0.043
|364.10
|5,666.72
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 2 H 1 tH 1 n 2
| rowspan="25" |[https://drive.google.com/drive/folders/18njhmOzRc67zCmVuLd0aDxl6ETBhL1gy?usp=sharing Google Drive]
|-
|811,301
|806,119
|0.64%
|0.159
|9.03
|1,413.42
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 3 H 1 tH 1 n 2
|-
|806,119
|736,690
|8.61%
|0.548
|35.21
|408.78
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 1 tH 1 n 2
|-
|736,690
|736,504
|0.03%
|0.009
|5.81
|23,021.56
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 1 H 1 tH 1 n 1
|-
|736,504
|735,317
|0.16%
|0.058
|5.71
|3,540.88
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 2 H 0 tH 0 n 2
|-
|735,317
|733,717
|0.22%
|0.341
|1.30
|599.28
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 2 tH 2 n 2
|-
|733,717
|673,920
|8.15%
|3.8
|4.43
|54.32
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 4 H 2 tH 2 n 4
|-
|673,920
|652,828
|3.13%
|~10
| ---
| ---
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 6 H 2 tH 2 n 4
|-
|652,828
|645,264
|1.16%
|~12
| ---
| ---
| style="text-align:left" |FAR CPS_LRU maxT 100000 LRUH 8 H 2 tH 2 n 4
|-
|645,264
|641,388
|0.60%
|~15
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 100000 LRUH 10 H 2 tH 2 n 10
|-
|641,388
|635,505
|0.92%
|~200
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 10 H 1 tH 2 n 10
|-
|635,505
|616,639
|2.97%
| ---
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 2 H 0 tH 0 n [3-10]
|-
|616,639
|592,039
|3.99%
|~700
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 3 H 0 tH 0 n [1-10]
|-
|592,039
|576,938
|2.55%
|~800
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 3 H [0-1] tH [0-1] n [1-10]
|-
|576,938
|572,963
|0.69%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4 H 0 tH 0 n [1-10]
|-
|572,963
|567,971
|0.87%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4 H 2 tH 0 n [1-10]
|-
|567,971
|566,096
|0.33%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 6 H 0 tH 0 n [1-10]
|-
|566,096
|564,290
|0.32%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 0 tH [0,2] n [1-10]
|-
|564,290
|559,553
|0.84%
|~1,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 2 tH 1 n [1-10]
|-
|559,553
|558,039
|0.27%
|~900
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 8 H 2 tH 2 n [1-10]
|-
|558,039
|556,814
|0.22%
|~14,000
| ---
| ---
|FAR CPS_LRU maxT 1000000 LRUH [12,16] H [0-2] tH [0-2] n [1-10]
|-
|556,814
|554,479
|0.42%
|~3,600
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH [1-3]
|-
|554,479
|551,586
|0.52%
|~5000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 4
|-
|551,586
|548,993
|0.47%
|~13,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 5
|-
|548,993
|545,005
|0.73%
|~57,000
| ---
| ---
|style="text-align:left"|FAR CPS_LRU maxT 1000000 LRUH 6 and 8
|}

== References ==


[[Category: BB Domains]][[Category:BB(2,6)]]

TMBR: February 2026

2026-03-30T20:04:52Z

ADucharme: copy editing

{{TMBRnav|January 2026|March 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

[[:Category:This Month in Beaver Research|This Month in Beaver Research]] for February 2026. This month, we had an excellent reduction in [[BB(6)]] holdouts, as well as a nice reduction in the [[BB(7)]] holdouts. Amazingly, we saw '''two''' [[BB(2,5)]] machines were proven nonhalting for the second month in a row. @LegionMammal978 created two new machines whose halting status is independent of the theories of [https://en.wikipedia.org/w/index.php?title=Peano_Arithmetic Peano Arithmetic] (372-state) and ZFC paired with the axiom "There exist arbitrarily large [[wikipedia:Subtle_cardinal|subtle cardinals]]" (493-state). For more context, see [[Logical independence]]. A new simulation method was introduced by @prurq - see [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 Discord]. Moreover, Tristan Stérin announced that the paper "Determination of the fifth Busy Beaver value" was accepted to the prestigious 58th ACM [[wikipedia:Symposium_on_Theory_of_Computing|Symposium on Theory of Computing]] ([https://acm-stoc.org/stoc2026/ STOC 2026]), and there would be a talk at the event in [[wikipedia:Salt_Lake_City|Salt Lake City]] in June 2026.

== Champions ==
* New champions were discovered for [[Busy Beaver for lambda calculus#Champions|BBλ(47)]] and BBλ(95). A [https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/laver.lam BBλ(201) champion] surpassing [https://en.wikipedia.org/wiki/Laver_table q(5)] was discovered by John Tromp, Bertram Felgenhauer, and 50_ft_lock.

== Misc ==

* @LegionMammal978 created two new nonhalting machines, whose halting status is independent of the theories of [https://en.wikipedia.org/w/index.php?title=Peano_Arithmetic Peano Arithmetic] (BB(372)) and ZFC+"There exist arbitrarily large [[wikipedia:Subtle_cardinal|subtle cardinals]]" (BB(493)) (see [[Logical independence]])
* Discord user prurq announced a new simulation method, "Cascade," which works especially well, see [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 Discord thread].
* @mxdys [https://discord.com/channels/960643023006490684/1226543091264126976/1469937272752177298 introduced a new longitudinal acceleration method], which [https://discord.com/channels/960643023006490684/1239205785913790465/1473950417275850804 had very fruitful results].

== Talks ==
* Tristan Stérin [https://discord.com/channels/960643023006490684/1151558585344593950/1467922688638062672 announced] that the paper "Determination of the fifth Busy Beaver value" was accepted for the 58th ACM [[wikipedia:Symposium_on_Theory_of_Computing|Symposium on Theory of Computing]] ([https://acm-stoc.org/stoc2026/ STOC 2026]), and there would be a talk at the event in [[wikipedia:Salt_Lake_City|Salt Lake City]] in June 2026

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(2,5)]]
|74
|72
|2
|2.70%
|-
|[[BB(6)]]
|1314
|1214
|100
|7.61%
|-
|[[BB(7)]]
|19,303,801
|18,195,192
|1,108,609
|5.74%
|-
|[[BB(2,6)]]
|558,039
|548,993
|9,046
|1.62%
|}
*[[BB(2,5)]]: '''2 solved machines.'''
**Andrew Ducharme found a machine nonhalting on [https://discord.com/channels/960643023006490684/1259770421046411285/1471227102844944510 11 Feb] via the mxdys C++ FAR decider. This was verified in Rocq by mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471228798505582602 the same day].
**mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471229409829847111 announced another TM proven the same day], which turned out to be a translated cycler.
**Peacemaker II [https://discord.com/channels/960643023006490684/1259770421046411285/1472647706835943596 determined the high-level behaviour of a holdout] could be transformed into a relatively simple-to-describe string rewriting problem.
*[[BB(6)]]: '''All''' '''machines simulated to 1e13''', '''100''' solved machines.
** prurq [https://discord.com/channels/960643023006490684/1239205785913790465/1471831607793946699 found a halting machine] with step count 30,505,241,149,212.
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1471837208615981179 followed up with 2 more halting machines the same day]. All 3 were verified in C++.
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1472051232746115173 found 7 non-halting machines] using the mxdys C++ FAR decider.
** Alistaire [https://discord.com/channels/960643023006490684/1239205785913790465/1472376779825090713 found a machine nonhalting] using Quick_Sim.py.
** prurq simulated 38 machines for >1e13 steps[https://discord.com/channels/960643023006490684/1471178503235043493/1471486886890704967 <nowiki>[19 machines]</nowiki>][https://docs.google.com/spreadsheets/d/1zMhtW_edMxrfUry-hVMFsDg3T1p_udjC2V2RKu6oSKE/edit?usp=sharing <nowiki>[19 more machines]</nowiki>] with his new method [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 "Cascade."]
** Alistaire [https://discord.com/channels/960643023006490684/1239205785913790465/1472246267345113158 simulated 13 machines] for >1e13 steps, 6 of which had already been simulated by prurq, essentially double-verifying them.
** Discord user @mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1472325414344069271 simulated a TM] for >1e13 steps, which also turned out to have been simulated by prurq already.
** For all machines simulated by prurq, see: [https://docs.google.com/spreadsheets/d/1zMhtW_edMxrfUry-hVMFsDg3T1p_udjC2V2RKu6oSKE/edit?gid=0#gid=0 Spreadsheet]. For all simulated by Alistaire (most machines), see: [https://discord.com/channels/960643023006490684/1239205785913790465/1472246267345113158 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1472376779825090713 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1472749954715095181 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1473010093284130847 <nowiki>[4]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1473035474745692351 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1473442168927686677 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1473785617208053900 <nowiki>[7]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1474120868761174040 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1474178147036434609 <nowiki>[9]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1474508316335144993 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1474685965409976561 <nowiki>[11]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1474903012446306354 <nowiki>[12]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475203181867958375 <nowiki>[13]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475272630478049371 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475564139438014554 <nowiki>[15]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475778581287403551 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475922318915272734 <nowiki>[17]</nowiki>]. '''All machines but 3 ([https://discord.com/channels/960643023006490684/1239205785913790465/1474508720330375460 <nowiki>[18]</nowiki>]) were ran to 1e13.'''
** Though announced on March 1st, prurq '''simulated the remaining machines to 1e13'''.[https://discord.com/channels/960643023006490684/1477591686514212894/1477716122617905305 <nowiki>[19]</nowiki>]
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1473950417275850804 released] a [[holdouts list]] of '''1226''' machines up to equivalence, some of which were decided via a [https://discord.com/channels/960643023006490684/1226543091264126976/1469937272752177298 new mxdys method for longitudinal acceleration].
** Andrew Ducharme found 9 non-halting machines in that list using the mxdys C++ FAR decider.[https://discord.com/channels/960643023006490684/1239205785913790465/1474302212284092436 <nowiki>[20]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475911180965904577 <nowiki>[21]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1477040884728987820 <nowiki>[22]</nowiki>]
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1477224991136419983 released] another holdouts list of '''1214''' machines up to equivalence.
** At the end of the month, the formal and Rocq-verified holdout counts are 1214, the informal holdout count is '''1212'''.
*[[BB(7)]]:
**Andrew Ducharme has reduced the number of holdouts from 19,303,801 to 18,254,545 (a 5.44% reduction) and then '''18,195,192''' (0.33%) using the mxdys C++ FAR decider.
*[[BB(2,6)]]:
**Andrew Ducharme continued reducing the number of holdouts, from 558,039 to '''551,586''' (a 1.16% reduction) using the mxdys C++ FAR decider.
**Another 0.47% reduction by Andrew Ducharme left '''548,993''' holdouts.[https://discord.com/channels/960643023006490684/1084047886494470185/1475216024734269644 <nowiki>[23]</nowiki>]

[[Category:This Month in Beaver Research|2026-02]]

TMBR: February 2026

2026-03-30T19:58:03Z

ADucharme: introduction edits + reorg

{{TMBRnav|January 2026|March 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).''

[[:Category:This Month in Beaver Research|This Month in Beaver Research]] for February 2026. This month, we had an excellent reduction in [[BB(6)]] holdouts, as well as a nice reduction in the [[BB(7)]] holdouts. Amazingly, we saw '''two''' [[BB(2,5)]] machines were proven nonhalting for the second month in a row. @LegionMammal978 created two new machines whose halting status is independent of the theories of [https://en.wikipedia.org/w/index.php?title=Peano_Arithmetic Peano Arithmetic] (372-state) and ZFC paired with the axiom "There exist arbitrarily large [[wikipedia:Subtle_cardinal|subtle cardinals]]" (493-state). For more context, see [[Logical independence]]. A new simulation method was introduced by @prurq - see [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 Discord]. Moreover, Tristan Stérin announced that the paper "Determination of the fifth Busy Beaver value" was accepted to the prestigious 58th ACM [[wikipedia:Symposium_on_Theory_of_Computing|Symposium on Theory of Computing]] ([https://acm-stoc.org/stoc2026/ STOC 2026]), and there would be a talk at the event in [[wikipedia:Salt_Lake_City|Salt Lake City]] in June 2026.

== Champions ==
* New champions were discovered for [[Busy Beaver for lambda calculus#Champions|BBλ(47)]] and BBλ(95). A [https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/laver.lam BBλ(201) champion] surpassing [https://en.wikipedia.org/wiki/Laver_table q(5)] was discovered by John Tromp, Bertram Felgenhauer, and 50_ft_lock.

== Misc ==

* @LegionMammal978 created two new nonhalting machines, whose halting status is independent of the theories of [https://en.wikipedia.org/w/index.php?title=Peano_Arithmetic Peano Arithmetic] (BB(372)) and ZFC+"There exist arbitrarily large [[wikipedia:Subtle_cardinal|subtle cardinals]]" (BB(493)) (see [[Logical independence]])
* Discord user prurq announced a new simulation method, "Cascade", which works especially well, see [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 Discord thread].
* @mxdys [https://discord.com/channels/960643023006490684/1226543091264126976/1469937272752177298 introduced a new longitudinal acceleration method], which [https://discord.com/channels/960643023006490684/1239205785913790465/1473950417275850804 had very fruitful results].

== Talks ==
* Tristan Stérin [https://discord.com/channels/960643023006490684/1151558585344593950/1467922688638062672 announced] that the paper "Determination of the fifth Busy Beaver value" was accepted for the 58th ACM [[wikipedia:Symposium_on_Theory_of_Computing|Symposium on Theory of Computing]] ([https://acm-stoc.org/stoc2026/ STOC 2026]), and there would be a talk at the event in [[wikipedia:Salt_Lake_City|Salt Lake City]] in June 2026

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(2,5)]]
|74
|72
|2
|2.70%
|-
|[[BB(6)]]
|1314
|1214
|100
|7.61%
|-
|[[BB(7)]]
|19,303,801
|18,195,192
|1,108,609
|5.74%
|-
|[[BB(2,6)]]
|558,039
|548,993
|9,046
|1.62%
|}
*[[BB(2,5)]]: '''2 solved machines.'''
**Andrew Ducharme found a machine nonhalting on [https://discord.com/channels/960643023006490684/1259770421046411285/1471227102844944510 11 Feb] via the mxdys C++ FAR decider. This was verified in Rocq by mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471228798505582602 the same day].
**mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471229409829847111 announced another TM proven the same day], which turned out to be a translated cycler.
**Peacemaker II [https://discord.com/channels/960643023006490684/1259770421046411285/1472647706835943596 found the high-level behaviour of a machine], which turned out to be a relatively simple-to-describe string rewriting problem of sorts.
*[[BB(6)]]: '''All''' '''machines simulated to 1e13''', '''100''' solved machines.
** prurq [https://discord.com/channels/960643023006490684/1239205785913790465/1471831607793946699 found a halting machine] with step count 30,505,241,149,212.
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1471837208615981179 followed up with 2 more halting machines the same day]. All 3 were verified in c++.
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1472051232746115173 found 7 non-halting machines] using the mxdys C++ FAR decider.
** Alistaire [https://discord.com/channels/960643023006490684/1239205785913790465/1472376779825090713 found a machine nonhalting] using Quick_Sim.py.
** prurq simulated 38 machines for >1e13 steps[https://discord.com/channels/960643023006490684/1471178503235043493/1471486886890704967 <nowiki>[19 machines]</nowiki>][https://docs.google.com/spreadsheets/d/1zMhtW_edMxrfUry-hVMFsDg3T1p_udjC2V2RKu6oSKE/edit?usp=sharing <nowiki>[19 more machines]</nowiki>] with his new method [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 "Cascade".]
** Alistaire [https://discord.com/channels/960643023006490684/1239205785913790465/1472246267345113158 simulated 13 machines] for >1e13 steps, 6 of which had already been simulated by prurq, essentialy double-verifying them.
** Discord user @mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1472325414344069271 simulated a TM] for >1e13 steps, which also turned out to have been simulated by prurq already.
** For all machines simulated by prurq, see: [https://docs.google.com/spreadsheets/d/1zMhtW_edMxrfUry-hVMFsDg3T1p_udjC2V2RKu6oSKE/edit?gid=0#gid=0 Spreadsheet], for all simulated by Alistaire (most machines), see: [https://discord.com/channels/960643023006490684/1239205785913790465/1472246267345113158 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1472376779825090713 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1472749954715095181 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1473010093284130847 <nowiki>[4]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1473035474745692351 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1473442168927686677 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1473785617208053900 <nowiki>[7]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1474120868761174040 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1474178147036434609 <nowiki>[9]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1474508316335144993 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1474685965409976561 <nowiki>[11]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1474903012446306354 <nowiki>[12]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475203181867958375 <nowiki>[13]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475272630478049371 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475564139438014554 <nowiki>[15]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475778581287403551 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475922318915272734 <nowiki>[17]</nowiki>]. '''All machines but 3: [https://discord.com/channels/960643023006490684/1239205785913790465/1474508720330375460 <nowiki>[18]</nowiki>] were ran to 1e13.'''
** Though on March 1st, most of the work was probably done in February: prurq '''simulated the remaining machines to 1e13'''.[https://discord.com/channels/960643023006490684/1477591686514212894/1477716122617905305 <nowiki>[19]</nowiki>]
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1473950417275850804 released] a [[holdouts list]] of '''1226''' machines up to equivalence, some of which were decided via [https://discord.com/channels/960643023006490684/1226543091264126976/1469937272752177298 new mxdys method for longitudinal acceleration].
** Andrew Ducharme found 9 non-halting machines in that list using the mxdys C++ FAR decider.[https://discord.com/channels/960643023006490684/1239205785913790465/1474302212284092436 <nowiki>[20]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475911180965904577 <nowiki>[21]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1477040884728987820 <nowiki>[22]</nowiki>]
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1477224991136419983 released] another holdouts list of '''1214''' machines up to equivalence.
** At the end of the month, the formal and Rocq-verified holdout counts are 1214, the informal holdout count is '''1212'''.
*[[BB(7)]]:
**Andrew Ducharme has reduced the number of holdouts from 19,303,801 to 18,254,545 (a 5.44% reduction) and then '''18,195,192''' (0.33%) using the mxdys C++ FAR decider.
*[[BB(2,6)]]:
**Andrew Ducharme continued reducing the number of holdouts, from 558,039 to '''551,586''' (a 1.16% reduction) using the mxdys C++ FAR decider.
**Another 0.47% reduction by Andrew Ducharme left '''548,993''' holdouts.[https://discord.com/channels/960643023006490684/1084047886494470185/1475216024734269644 <nowiki>[23]</nowiki>]

[[Category:This Month in Beaver Research|2026-02]]

BB(4,3)

2026-03-17T20:21:23Z

ADucharme: /* Top Halters */ TMify

The Busy Beaver problem for 4 states and 3 symbols is unsolved. The existence of [[Cryptids]] in the domain is given by the discovery of [[Bigfoot]] in [[BB(3,3)]]. The current [[Champions#3-Symbol TMs|champion]] is {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} which was discovered by Pavel Kropitz in May 2024 along with 6 other long running machines. It was [[User:Polygon/Page for analyses#1RB1RD1LC 2LB1RB1LC 1RZ1LA1LD 0RB2RA2RD (bbch)|analyzed by Polygon]] in Oct 2025, demonstrating the lower bounds:

<math display="block">S(4,3) > \Sigma(4,3) > 10 \uparrow^{4} 4</math>

== Top Halters ==
The longest running halting BB(4,3) TMs are split amongst two classes: the pentational and hexational TMs found by Pavel Kropitz outlined in the Potential Champions section, and the tetrational TMs found by comprehensive holdout filtering by Terry Ligocki. 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 longest running halters found by Pavel Kropitz are:
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|}
The top 20 scoring halting machines found by comprehensive search are:
{| class="wikitable"
|+
!Standard format
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB0LC1RC_1LA2RB1LB_1RC2LA0RD_2LB1RZ2LC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LA1RA_1LA0RC1LC_1LC2RB0LD_2RA1RZ2RC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA1LB_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB0RB1LC_2LC0LD1RA_2RB2LD1RZ_2LA2LB0LD}}
|~10 ↑↑ 141.44248
|Andrew Ducharme
|-
|{{TM|1RB1RC2RB_2LC2LB0LC_1RA1LD0RB_2RA0LC1RZ|halt}}
|~10 ↑↑ 128.27662
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2RA2RB_1LD2LA0RC_1RA1RZ0RB|halt}}
|~10 ↑↑ 127.14811
|Terry Ligocki
|-
|{{TM|1RB1RC2LA_2LC0LA1LD_0LD0LB1RZ_2RA2RD1LB|halt}}
|~10 ↑↑ 107.56135
|Terry Ligocki
|-
|{{TM|1RB2LB1LA_0LC2RA0RA_2LA2LD1RZ_2LB2LC2LC|halt}}
|~10 ↑↑ 86.27662
|Terry Ligocki
|-
|{{TM|1RB1LD1RC_0LC0RB0LD_2RA1LA1RZ_2LC2LB0LA|halt}}
|~10 ↑↑ 86.15130
|Terry Ligocki
|-
|{{TM|1RB0LD0RC_1RC1RZ0RB_1LA2RD2RC_2LD2LB0RD|halt}}
|~10 ↑↑ 85.27623
|Terry Ligocki
|-
|{{TM|1RB0LD0RC_2LC2LB1RZ_2LD0LC1RD_1RA1LA0LA}}
|~10 ↑↑ 83.24824
|Andrew Ducharme
|-
|{{TM|1RB2RB1LC_2LB2RA0RB_0LC1RZ2LD_2LA2RA0LB|halt}}
|~10 ↑↑ 83.00625
|Terry Ligocki
|-
|{{TM|1RB2RB1LC_2LB2RA0RB_0RC1RZ2LD_2LA2RA0LB|halt}}
|~10 ↑↑ 83.00625
|Terry Ligocki
|-
|{{TM|1RB2RB1LC_2LB2RA0RB_1LC1RZ2LD_2LA2RA0LB|halt}}
|~10 ↑↑ 83.00625
|Terry Ligocki
|-
|{{TM|1RB2RB1LC_2LB2RA0RB_1RC1RZ2LD_2LA2RA0LB|halt}}
|~10 ↑↑ 83.00625
|Terry Ligocki
|-
|{{TM|1RB2RB1LC_2LB2RA0RB_1RZ---2LD_2LA2RA0LB|halt}}
|~10 ↑↑ 83.00625
|Terry Ligocki
|}

== Potential Champions ==
In May 2024, [https://discord.com/channels/960643023006490684/1026577255754903572/1243253180297646120 Pavel Kropitz found 7 halting TMs] that run for a large number of steps. Four of these are equivalent and were [https://discord.com/channels/960643023006490684/1331570843829932063/1337228898068463718 analyzed by Racheline] in February 2025, while the remaining three were [[User:Polygon/Page for analyses|analyzed by Polygon in October 2025.]]
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}*
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}
|<math>10 \uparrow\uparrow 9.873987</math>
|}
<nowiki>*</nowiki>equivalent to {{TM|0RB1RZ1RC_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}, {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ0RA|halt}} and {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ1RB|halt}}.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in December 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~633B(illion) TMs of which ~34.4B TMs were holdouts. Also found were ~206B halting TMs and ~392B infinite TMs. The number of holdouts was reduced to ~461M TMs (a 98.66% reduction).

Two C++ programs were run before the filters in the table.
<pre>
lr_enum 4 3 8 /dev/null /dev/null 4x3.unk.txt false
00 <= XX < 47: lr_enum_continue 4x3.in.XX 1000 /dev/null /dev/null 4x3.unk.txt.XX XX false
</pre>
Both do the initial enumeration and simple filtering. The "/dev/null" in both commands would be files where the halting and infinite TMs would be stored. The first command generates the TMs from a TNF tree for BB(4,3) of depth 8 and outputs the holdouts to 4x3.unk.txt. This file was then divided into 48 pieces, 4x3.in.XX, 0 <= XX < 47. The second commands (one for each XX) continues the enumeration by running each TM for 1,000 steps. It classifies each as halting, infinite, or unknown/holdout. Again, the halting and infinite TMs are "written" to /dev/null, i.e., they aren't saved. The holdouts are stored in 48 files: 4x3.unk.txt.XX.

For these runs the first command generated a total of ~45M TMs: ~1.86M halting, ~774K infinite, and ~42.0M holdouts. The second took the ~42.0M holdout TMs and generated a total of ~633B TMs: ~206B halting, ~392B infinite, and ~34.4B holdouts. These holdouts were used as a starting point of the filters below.

The "Description" column in the table below contain the command run. Two options are not given, "--infile=..." and an "--outfile=...". These are necessary and specify where to read and write the results, respectively. Note: The work flow was to divide the input holdouts into 48 pieces, run the command on each piece simultaneously on one of 48 cores, and then combine the 48 results into a group of holdouts.

The details are given in this table:

(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"
!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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|34,413,860,527
|30,874,934,791
|10.28%
|646.6
|1,520.36
|14,784.57
|style="text-align:left" |Reverse_Engineer_Filter.py
|rowspan="10" style="text-align:left" |[https://drive.google.com/drive/folders/1KMOVgngtUVMEA7EjxtNcsgksQ5Y4tby9?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|30,874,934,791
|12,942,386,396
|58.08%
|4,134.8
|1,204.72
|2,074.19
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|12,942,386,396
|4,534,322,415
|64.97%
|3,361.1
|694.88
|1,069.62
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|4,534,322,415
|2,959,598,830
|34.73%
|3,318.1
|131.83
|379.59
|style="text-align:left" |CPS_Filter.py --block-size=3
|-
|style="text-align:left" |Terry Ligocki
|2,959,598,830
|1,651,940,618
|44.18%
|2,700.6
|134.50
|304.42
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --no-steps --time=0.002 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|1,651,940,618
|854,984,279
|48.24%
|2,276.3
|97.25
|201.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.005 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|854,984,279
|683,163,325
|20.10%
|430.1
|110.96
|552.15
|style="text-align:left" |CPS_Filter.py --block-size=4 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|683,163,325
|460,916,384
|32.53%
|5,507.9
|11.21
|34.45
|style="text-align:left" |CPS_Filter.py --min-block-size=1 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:center" |'''Cumulative'''
|'''632,656,365,801'''
|'''460,916,384'''
|'''98.66%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

== Phase 2 ==

When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===

Starting from the results of Phase 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 33 variations were run. The holdouts were reduced from ~461B TMs to ~33.9M TMs (a 92.7% reduction). The details are given in the table below, including links to the Google Drive with the holdouts. Entries with multiple lines represent runs where all the commands in the "Description" were applied during one run.

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|460,916,384
|234,834,703
|49.05%
|96.7
|649.48
|1,324.10
|style="text-align:left" | chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 1000 NG_n 2 run
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1tFtg1eFC-AdqCzh7XNmx5O2mTQwtaNbm?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|234,834,703
|160,518,206
|31.65%
|70.9
|291.33
|920.57
|style="text-align:left" | chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 1000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|160,518,206
|132,296,033
|17.58%
|41.5
|188.86
|1,074.17
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 6 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|132,296,033
|113,193,595
|14.44%
|54.9
|96.57
|668.77
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|113,193,595
|85,920,795
|24.09%
|106.8
|70.96
|294.52
|style="text-align:left" | chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|85,920,795
|78,674,774
|8.43%
|28.9
|69.62
|825.51
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|78,674,774
|73,228,547
|6.92%
|68.7
|22.02
|318.04
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|73,228,547
|67,014,897
|8.49%
|23.2
|74.50
|878.02
|style="text-align:left" | chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|67,014,897
|57,625,231
|14.01%
|75.6
|34.49
|246.13
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|57,625,231
|48,070,606
|16.58%
|645.4
|4.11
|24.80
|style="text-align:left" | chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 30000 NG_n 10 run
|-
|style="text-align:left" |Terry Ligocki
|48,070,606
|44,254,286
|7.94%
|166.3
|6.38
|80.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 6 H 1 tH 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|44,254,286
|40,836,159
|7.72%
|188.3
|5.04
|65.29
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|40,836,159
|37,460,692
|8.27%
|192.3
|4.88
|58.99
|style="text-align:left" |
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 2 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 3 n 2 run 
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 3000 NG_n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 4 H 1 tH 1 n 2 run 
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 3 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 2 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|37,460,692
|36,167,570
|3.45%
|237.7
|1.51
|43.77
|style="text-align:left" |
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 3 H 0 tH 1 n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 10 chr_H 8 MitM_CTL NG maxT 10000 NG_n 4 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 2 run 
|-
|style="text-align:left" |Terry Ligocki
|36,167,570
|34,642,544
|4.22%
|467.2
|0.91
|21.50
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,642,544
|34,339,943
|0.87%
|383.1
|0.22
|25.12
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 0 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,339,943
|33,860,069
|1.40%
|666.5
|0.20
|14.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 2 tH 2 n 8 run
|-
|style="text-align:center" |'''Cumulative'''
|'''460,916,384'''
|'''33,860,069'''
|'''92.70%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

=== Stage 2 ===

Starting from the results of Phase 2 Stage, Terry Ligocki ran a variety of enumeration and decider codes. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~33.9M TMs to ~9.4M TMs (a 72.2% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|33,860,069
|21,065,769
|37.79%
|93.0
|38.20
|101.11
|style="text-align:left" |lr_enum_continue 4x3.in.txt 1000000 4x3.halt.txt 4x3.inf.txt 4x3.holdouts.txt 00 false
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|21,065,769
|18,949,009
|10.05%
|5,566.1
|0.11
|1.05
|style="text-align:left" |Enumerate.py max-loops 100_000 block-size 2 --tape-limit 1_000 --no-steps --time 1.0 --recursive --exp-linear-rules --lin-steps 0 --no-ctl --no-reverse-engineer --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,949,009
|18,138,027
|4,28%
|0.4
|511.59
|11953.46
|style="text-align:left" |Reverse_Engineer_Filter.py --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,138,027
|11,985,999
|33.92%
|4.8
|352.73
|1,039.95
|style="text-align:left" | chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|11,985,999
|9,988,715
|16.66%
|640.4
|0.87
|5.20
|style="text-align:left" |
chr_LRUH 24 chr_H 16 MitM_CTL NG maxT 30000 NG_n 3 run 
chr_LRUH 14 chr_H 2 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 5 run 
chr_asth 0 chr_LRUH 48 chr_H 48 MitM_CTL NG maxT 30000 NG_n 5 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 3 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 1 n 4 run 
chr_LRUH 14 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 0 n 6 run 
chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 18 chr_H 16 MitM_CTL NG maxT 30000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 1 tH 0 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 3 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|9,988,715
|9,401,447
|5.88%
|1,398.7
|0.12
|1.98
|style="text-align:left" |
chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 100000 NG_n 5 run 
chr_LRUH 22 chr_H 12 MitM_CTL NG maxT 100000 NG_n 6 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 100000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 16 H 1 tH 0 n 10 run 
chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 1000000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 6 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 2 tH 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 6 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 1 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 5 n 2 run 
|-
|style="text-align:center" |'''Cumulative'''
|'''33,860,069'''
|'''9,401,447'''
|'''72.23%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

==References==

[[Category:BB Domains]][[Category:BB(4,3)]]

BB(4,3)

2026-03-17T20:20:52Z

ADucharme: /* Top Halters */ add discoverer column, two new top 20 halters

The Busy Beaver problem for 4 states and 3 symbols is unsolved. The existence of [[Cryptids]] in the domain is given by the discovery of [[Bigfoot]] in [[BB(3,3)]]. The current [[Champions#3-Symbol TMs|champion]] is {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} which was discovered by Pavel Kropitz in May 2024 along with 6 other long running machines. It was [[User:Polygon/Page for analyses#1RB1RD1LC 2LB1RB1LC 1RZ1LA1LD 0RB2RA2RD (bbch)|analyzed by Polygon]] in Oct 2025, demonstrating the lower bounds:

<math display="block">S(4,3) > \Sigma(4,3) > 10 \uparrow^{4} 4</math>

== Top Halters ==
The longest running halting BB(4,3) TMs are split amongst two classes: the pentational and hexational TMs found by Pavel Kropitz outlined in the Potential Champions section, and the tetrational TMs found by comprehensive holdout filtering by Terry Ligocki. 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 longest running halters found by Pavel Kropitz are:
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|}
The top 20 scoring halting machines found by comprehensive search are:
{| class="wikitable"
|+
!Standard format
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB0LC1RC_1LA2RB1LB_1RC2LA0RD_2LB1RZ2LC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LA1RA_1LA0RC1LC_1LC2RB0LD_2RA1RZ2RC|halt}}
|~10 ↑↑ 190.21359
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA0RD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD0RC2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA1LB_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2LD2RB_2LD2LA2LD_2LA1RZ0RD|halt}}
|~10 ↑↑ 166.03664
|Terry Ligocki
|-
|1RB0RB1LC_2LC0LD1RA_2RB2LD1RZ_2LA2LB0LD
|~10 ↑↑ 141.44248
|Andrew Ducharme
|-
|{{TM|1RB1RC2RB_2LC2LB0LC_1RA1LD0RB_2RA0LC1RZ|halt}}
|~10 ↑↑ 128.27662
|Terry Ligocki
|-
|{{TM|1RB2LC1RA_1LA2RA2RB_1LD2LA0RC_1RA1RZ0RB|halt}}
|~10 ↑↑ 127.14811
|Terry Ligocki
|-
|{{TM|1RB1RC2LA_2LC0LA1LD_0LD0LB1RZ_2RA2RD1LB|halt}}
|~10 ↑↑ 107.56135
|Terry Ligocki
|-
|{{TM|1RB2LB1LA_0LC2RA0RA_2LA2LD1RZ_2LB2LC2LC|halt}}
|~10 ↑↑ 86.27662
|Terry Ligocki
|-
|{{TM|1RB1LD1RC_0LC0RB0LD_2RA1LA1RZ_2LC2LB0LA|halt}}
|~10 ↑↑ 86.15130
|Terry Ligocki
|-
|{{TM|1RB0LD0RC_1RC1RZ0RB_1LA2RD2RC_2LD2LB0RD|halt}}
|~10 ↑↑ 85.27623
|Terry Ligocki
|-
|1RB0LD0RC_2LC2LB1RZ_2LD0LC1RD_1RA1LA0LA
|~10 ↑↑ 83.24824
|Andrew Ducharme
|-
|{{TM|1RB2RB1LC_2LB2RA0RB_0LC1RZ2LD_2LA2RA0LB|halt}}
|~10 ↑↑ 83.00625
|Terry Ligocki
|-
|{{TM|1RB2RB1LC_2LB2RA0RB_0RC1RZ2LD_2LA2RA0LB|halt}}
|~10 ↑↑ 83.00625
|Terry Ligocki
|-
|{{TM|1RB2RB1LC_2LB2RA0RB_1LC1RZ2LD_2LA2RA0LB|halt}}
|~10 ↑↑ 83.00625
|Terry Ligocki
|-
|{{TM|1RB2RB1LC_2LB2RA0RB_1RC1RZ2LD_2LA2RA0LB|halt}}
|~10 ↑↑ 83.00625
|Terry Ligocki
|-
|{{TM|1RB2RB1LC_2LB2RA0RB_1RZ---2LD_2LA2RA0LB|halt}}
|~10 ↑↑ 83.00625
|Terry Ligocki
|}

== Potential Champions ==
In May 2024, [https://discord.com/channels/960643023006490684/1026577255754903572/1243253180297646120 Pavel Kropitz found 7 halting TMs] that run for a large number of steps. Four of these are equivalent and were [https://discord.com/channels/960643023006490684/1331570843829932063/1337228898068463718 analyzed by Racheline] in February 2025, while the remaining three were [[User:Polygon/Page for analyses|analyzed by Polygon in October 2025.]]
{| class="wikitable"
!Standard format
!Approximate sigma scores
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}
|<math>10 \uparrow^{4} 4</math>
|-
|{{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}*
|<math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>
|-
|{{TM|1RB2LB0LB_2LC2LA0LA_2RD1LC1RZ_1RA2LD1RD|halt}}
|<math>3 \uparrow\uparrow\uparrow 88574</math>
|-
|{{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_2RB2RA2RD|halt}}
|<math>10 \uparrow\uparrow 9.873987</math>
|}
<nowiki>*</nowiki>equivalent to {{TM|0RB1RZ1RC_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}}, {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ0RA|halt}} and {{TM|1RB1LA2LA_1LA2RC1LB_1RD2RB0LC_0RA1RZ1RB|halt}}.

== Phase 1 ==
The initial phase of enumeration and reduction of [[holdouts]] took place in December 2024 and was done by Terry Ligocki using the Ligockis' C++ and Python codes. The initial enumerations generated ~633B(illion) TMs of which ~34.4B TMs were holdouts. Also found were ~206B halting TMs and ~392B infinite TMs. The number of holdouts was reduced to ~461M TMs (a 98.66% reduction).

Two C++ programs were run before the filters in the table.
<pre>
lr_enum 4 3 8 /dev/null /dev/null 4x3.unk.txt false
00 <= XX < 47: lr_enum_continue 4x3.in.XX 1000 /dev/null /dev/null 4x3.unk.txt.XX XX false
</pre>
Both do the initial enumeration and simple filtering. The "/dev/null" in both commands would be files where the halting and infinite TMs would be stored. The first command generates the TMs from a TNF tree for BB(4,3) of depth 8 and outputs the holdouts to 4x3.unk.txt. This file was then divided into 48 pieces, 4x3.in.XX, 0 <= XX < 47. The second commands (one for each XX) continues the enumeration by running each TM for 1,000 steps. It classifies each as halting, infinite, or unknown/holdout. Again, the halting and infinite TMs are "written" to /dev/null, i.e., they aren't saved. The holdouts are stored in 48 files: 4x3.unk.txt.XX.

For these runs the first command generated a total of ~45M TMs: ~1.86M halting, ~774K infinite, and ~42.0M holdouts. The second took the ~42.0M holdout TMs and generated a total of ~633B TMs: ~206B halting, ~392B infinite, and ~34.4B holdouts. These holdouts were used as a starting point of the filters below.

The "Description" column in the table below contain the command run. Two options are not given, "--infile=..." and an "--outfile=...". These are necessary and specify where to read and write the results, respectively. Note: The work flow was to divide the input holdouts into 48 pieces, run the command on each piece simultaneously on one of 48 cores, and then combine the 48 results into a group of holdouts.

The details are given in this table:

(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"
!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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|34,413,860,527
|30,874,934,791
|10.28%
|646.6
|1,520.36
|14,784.57
|style="text-align:left" |Reverse_Engineer_Filter.py
|rowspan="10" style="text-align:left" |[https://drive.google.com/drive/folders/1KMOVgngtUVMEA7EjxtNcsgksQ5Y4tby9?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|30,874,934,791
|12,942,386,396
|58.08%
|4,134.8
|1,204.72
|2,074.19
|style="text-align:left" |CPS_Filter.py --block-size=1
|-
|style="text-align:left" |Terry Ligocki
|12,942,386,396
|4,534,322,415
|64.97%
|3,361.1
|694.88
|1,069.62
|style="text-align:left" |CPS_Filter.py --block-size=2
|-
|style="text-align:left" |Terry Ligocki
|4,534,322,415
|2,959,598,830
|34.73%
|3,318.1
|131.83
|379.59
|style="text-align:left" |CPS_Filter.py --block-size=3
|-
|style="text-align:left" |Terry Ligocki
|2,959,598,830
|1,651,940,618
|44.18%
|2,700.6
|134.50
|304.42
|style="text-align:left" |Enumerate.py --max-loops=1_000 --block-size=2 --no-steps --time=0.002 --lin-steps=0 --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|1,651,940,618
|854,984,279
|48.24%
|2,276.3
|97.25
|201.59
|style="text-align:left" |Enumerate.py --max-loops=10_000 --block-size=12 --no-steps --time=0.005 --lin-steps=0 --no-ctl --no-reverse-engineer --save-freq=10_000
|-
|style="text-align:left" |Terry Ligocki
|854,984,279
|683,163,325
|20.10%
|430.1
|110.96
|552.15
|style="text-align:left" |CPS_Filter.py --block-size=4 --max-steps=1_000
|-
|style="text-align:left" |Terry Ligocki
|683,163,325
|460,916,384
|32.53%
|5,507.9
|11.21
|34.45
|style="text-align:left" |CPS_Filter.py --min-block-size=1 --max-block-size=6 --max-steps=10_000
|-
|style="text-align:center" |'''Cumulative'''
|'''632,656,365,801'''
|'''460,916,384'''
|'''98.66%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

== Phase 2 ==

When Phase 1 was completed, a set of deciders/parameters were run to reduce the number of holdout TMs. The details are given in the various Stages below.

=== Stage 1 ===

Starting from the results of Phase 1, Terry Ligocki ran @mxdys' C++ code, "main.exe", using a variety of its deciders with various parameters. A total of 33 variations were run. The holdouts were reduced from ~461B TMs to ~33.9M TMs (a 92.7% reduction). The details are given in the table below, including links to the Google Drive with the holdouts. Entries with multiple lines represent runs where all the commands in the "Description" were applied during one run.

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|460,916,384
|234,834,703
|49.05%
|96.7
|649.48
|1,324.10
|style="text-align:left" | chr_LRUH 4 chr_H 2 MitM_CTL NG maxT 1000 NG_n 2 run
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1tFtg1eFC-AdqCzh7XNmx5O2mTQwtaNbm?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|234,834,703
|160,518,206
|31.65%
|70.9
|291.33
|920.57
|style="text-align:left" | chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 1000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|160,518,206
|132,296,033
|17.58%
|41.5
|188.86
|1,074.17
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 6 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|132,296,033
|113,193,595
|14.44%
|54.9
|96.57
|668.77
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 4 mod 1 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|113,193,595
|85,920,795
|24.09%
|106.8
|70.96
|294.52
|style="text-align:left" | chr_LRUH 16 chr_H 12 MitM_CTL NG maxT 3000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|85,920,795
|78,674,774
|8.43%
|28.9
|69.62
|825.51
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|78,674,774
|73,228,547
|6.92%
|68.7
|22.02
|318.04
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|73,228,547
|67,014,897
|8.49%
|23.2
|74.50
|878.02
|style="text-align:left" | chr_LRUH 4 chr_H 4 MitM_CTL NG maxT 30000 NG_n 1 run
|-
|style="text-align:left" |Terry Ligocki
|67,014,897
|57,625,231
|14.01%
|75.6
|34.49
|246.13
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 3000 H 4 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|57,625,231
|48,070,606
|16.58%
|645.4
|4.11
|24.80
|style="text-align:left" | chr_LRUH 18 chr_H 12 MitM_CTL NG maxT 30000 NG_n 10 run
|-
|style="text-align:left" |Terry Ligocki
|48,070,606
|44,254,286
|7.94%
|166.3
|6.38
|80.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 6 H 1 tH 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|44,254,286
|40,836,159
|7.72%
|188.3
|5.04
|65.29
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 1 n 2 run
|-
|style="text-align:left" |Terry Ligocki
|40,836,159
|37,460,692
|8.27%
|192.3
|4.88
|58.99
|style="text-align:left" |
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 2 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 1000 LRUH 6 H 0 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 3 n 2 run 
chr_LRUH 6 chr_H 4 MitM_CTL NG maxT 3000 NG_n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 3000 LRUH 4 H 1 tH 1 n 2 run 
chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 3000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 3 mod 3 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 1000 H 8 mod 2 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 2 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|37,460,692
|36,167,570
|3.45%
|237.7
|1.51
|43.77
|style="text-align:left" |
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 3 H 0 tH 1 n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run 
chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 6 chr_H 6 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 10 chr_H 8 MitM_CTL NG maxT 10000 NG_n 4 run 
MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 2 run 
|-
|style="text-align:left" |Terry Ligocki
|36,167,570
|34,642,544
|4.22%
|467.2
|0.91
|21.50
|style="text-align:left" | MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,642,544
|34,339,943
|0.87%
|383.1
|0.22
|25.12
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 1 tH 0 n 24 run
|-
|style="text-align:left" |Terry Ligocki
|34,339,943
|33,860,069
|1.40%
|666.5
|0.20
|14.31
|style="text-align:left" | MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 12 H 2 tH 2 n 8 run
|-
|style="text-align:center" |'''Cumulative'''
|'''460,916,384'''
|'''33,860,069'''
|'''92.70%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

=== Stage 2 ===

Starting from the results of Phase 2 Stage, Terry Ligocki ran a variety of enumeration and decider codes. Some of these runs generated new TMs due to the BB(4,3) TNF tree not being fully generated at this time. These reduced the number of holdouts from ~33.9M TMs to ~9.4M TMs (a 72.2% reduction). The details are given in the table below, including links to the Google Drive with the holdouts, halting, and infinite TMs:

(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" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|33,860,069
|21,065,769
|37.79%
|93.0
|38.20
|101.11
|style="text-align:left" |lr_enum_continue 4x3.in.txt 1000000 4x3.halt.txt 4x3.inf.txt 4x3.holdouts.txt 00 false
|rowspan="20" style="text-align:left" |[https://drive.google.com/drive/folders/1qNssnvK3W2jJ68VBq9FJZMy9TvwbQk4_?usp=drive_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|21,065,769
|18,949,009
|10.05%
|5,566.1
|0.11
|1.05
|style="text-align:left" |Enumerate.py max-loops 100_000 block-size 2 --tape-limit 1_000 --no-steps --time 1.0 --recursive --exp-linear-rules --lin-steps 0 --no-ctl --no-reverse-engineer --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,949,009
|18,138,027
|4,28%
|0.4
|511.59
|11953.46
|style="text-align:left" |Reverse_Engineer_Filter.py --infile 4x3.in.txt --outfile 4x3.out.pb
|-
|style="text-align:left" |Terry Ligocki
|18,138,027
|11,985,999
|33.92%
|4.8
|352.73
|1,039.95
|style="text-align:left" | chr_asth 0 chr_LRUH 1 chr_H 1 MitM_CTL NG maxT 100000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|11,985,999
|9,988,715
|16.66%
|640.4
|0.87
|5.20
|style="text-align:left" |
chr_LRUH 24 chr_H 16 MitM_CTL NG maxT 30000 NG_n 3 run 
chr_LRUH 14 chr_H 2 MitM_CTL NG maxT 10000 NG_n 4 run 
chr_LRUH 2 chr_H 2 MitM_CTL NG maxT 3000 NG_n 5 run 
chr_asth 0 chr_LRUH 48 chr_H 48 MitM_CTL NG maxT 30000 NG_n 5 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 6 mod 3 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 1 tH 1 n 4 run 
chr_LRUH 14 chr_H 8 MitM_CTL NG maxT 10000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 0 n 6 run 
chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 30000 NG_n 2 run 
chr_LRUH 18 chr_H 16 MitM_CTL NG maxT 30000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 3 H 1 tH 0 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 100000 H 3 mod 3 n 1 run 
|-
|style="text-align:left" |Terry Ligocki
|9,988,715
|9,401,447
|5.88%
|1,398.7
|0.12
|1.98
|style="text-align:left" |
chr_asth 0 chr_LRUH 60 chr_H 60 MitM_CTL NG maxT 100000 NG_n 5 run 
chr_LRUH 22 chr_H 12 MitM_CTL NG maxT 100000 NG_n 6 run 
chr_LRUH 12 chr_H 12 MitM_CTL NG maxT 100000 NG_n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 16 H 1 tH 0 n 10 run 
chr_LRUH 4 chr_H 0 MitM_CTL NG maxT 1000000 NG_n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 6 n 1 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 3 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 2 run 
MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 8 H 2 tH 2 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 3 mod 2 n 3 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 6 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 30000 H 4 mod 2 n 1 run 
MitM_CTL CPS_LRU sim 1001 maxT 100000 LRUH 4 H 1 tH 1 n 2 run 
MitM_CTL RWL_mod sim 1001 maxT 10000 H 4 mod 5 n 2 run 
|-
|style="text-align:center" |'''Cumulative'''
|'''33,860,069'''
|'''9,401,447'''
|'''72.23%'''
| ---
| ---
| ---
|style="text-align:left" | ---
|}

==References==

[[Category:BB Domains]][[Category:BB(4,3)]]

BB(2,5)

2026-03-16T16:13:22Z

ADucharme: /* Holdouts */ move 2 formally solved TMs out of informal proof section

The 2-state, 5-symbol Busy Beaver problem, '''BB(2,5)''', is unsolved. With the discovery of the [[Cryptids|Cryptid]] machine [[Hydra]] in April 2024, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(2,5) and thus [https://www.sligocki.com/2024/05/10/bb-2-5-is-hard.html BB(2,5) is Hard].

The current BB(2,5) [[Champions#5-Symbol TMs|champion]] {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}} was discovered by Daniel Yuan in June 2024, proving the lower bounds:

<math display="block">S(2,5) > \Sigma(2,5) > 10^{10^{10^{3\,314\,360}}} > 10 \uparrow\uparrow 4</math>

== Cryptids ==
Known Cryptids:

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA}}, known as [[Hydra]]
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}, known as the [[Bonus Cryptid]]
Potential Cryptids:

* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}. [https://discord.com/channels/960643023006490684/1354037062830919690/1354037062830919690 Shift overflow counter]
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}.
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}.
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}}. [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]
* {{TM|1RB2RA3LB---2LB_2LA0LA4RB0RB1LA}}. Probviously halting. 1/8 chance of beating champ.

==Top Halters==
The 20 longest running known halting BB(2,5) TMs are:
{| class="wikitable"
|+
!Standard format
!(approximate) runtime
!Discoverer
|-
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|<math>10 \uparrow\uparrow 4.8142742</math>
|Daniel Yuan
|-
|{{TM|1RB2LB4LB3LA1RZ_1LA3RA3LB0LB0RA|halt}}
|<math>>10^{38\,033}</math>
|Pavel Kropitz
|-
|{{TM|1RB2LA1RA2LB2LA_0LA2RB3RB4RA1RZ|halt}}
|<math>>1.9 \times 10^{704}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2RA3LA4RB---_2LA3RB3RA1LB3LB|halt}}
|<math>>8.3 \times 10^{466}</math> (lower bound given by score)
|Daniel Yuan
|-
|{{TM|1RB2LA4RA2LB2LA_0LA2RB3RB1RA1RZ|halt}}
|<math>>1.6 \times 10^{211}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2LA4RA2LB2LA_0LA2RB3RB4RA1RZ|halt}}
|<math>>1.6 \times 10^{211}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2LA4RA1LB2LA_0LA2RB3RB2RA1RZ|halt}}
|<math>>5.2 \times 10^{61}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB0RB4RA2LB2LA_2LA1LB3RB4RA1RZ|halt}}
|<math>>7 \times 10^{21}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB1RZ4LA4LB2RA_2LB2RB3RB2RA0RB|halt}}
|<math>>9 \times 10^{16}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB3LA1LA0LB1RA_2LA4LB4LA1RA1RZ|halt}}
|<math>>3.77 \times 10^{16}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2RA1LA3LA2RA_2LA3RB4LA1LB1RZ|halt}}
|<math>>9 \times 10^{15}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2RA1LA1LB3LB_2LA3RB1RZ4RA1LA|halt}}
|417,310,842,648,366
|Terry and Shawn Ligocki
|-
|{{TM|1RB3LA1LA4LA1RA_2LB2RA1RZ0RA0RB|halt}}
|26,375,397,569,930
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB3LB4LB4LA2RA_2LA1RZ3RB4RA3RB|halt}}
|14,103,258,269,249
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB3RA4LB2RA3LA_2LA1RZ4RB4RB2LB|halt}}
|3,793,261,759,791
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB3RA1LA1LB3LB_2LA4LB3RA2RB1RZ|halt}}
|924,180,005,181
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB3LB1RZ1LA1LA_2LA3RB4LB4LB3RA|halt}}
|912,594,733,606
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB2RB3LA2RA3RA_2LB2LA3LA4RB1RZ|halt}}
|469,121,946,086
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB3RB3RB1LA3LB_2LA3RA4LB2RA1RZ|halt}}
|233,431,192,481
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB3LA1LB1RA3RA_2LB3LA3RA4RB1RZ|halt}}
|8,619,024,596
|Grégory Lafitte and Christophe Papazian
|}

== Certified progress ==
In April 2024, Shawn Ligocki publicly released a list of 23,411 undecided BB(2,5) machines. Justin Blanchard then made substantial progress over the course of the next month, reducing the list to 499 [[Holdouts lists|holdouts]] by late May 2024. In June 2024, @mxdys cut down the list to 273 using halting and inductive deciders, and again to 217 using [[Closed Tape Language|CTL]]. In February 2025, @mxdys ran a decider pipeline in Rocq that resulted in only 173 holdouts. Since then, additional machines have been proven in Rocq using both deciders and individual proofs.

On [https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 29 Mar 2025], @mxdys published a list of 83 holdouts that withstood state-of-the-art Rocq deciders.

Over the course of 5 months, @mxdys added 8 machines to Rocq[https://discord.com/channels/960643023006490684/1259770421046411285/1355799763437752521 1][https://discord.com/channels/960643023006490684/1259770421046411285/1355828077023854752 2][https://discord.com/channels/960643023006490684/1259770421046411285/1379521528869421137 3][https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 4][https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 5], lowering the certified holdout count to 75. There are 11 informal arguments, lowering the informal holdout count to 64.

Then, on [https://discord.com/channels/960643023006490684/1259770421046411285/1466208979511414885 29 Jan 2026], Andrew Ducharme found a machine nonhalting. This was verified by @mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1466331107279769736 the same day]. Hence the certified holdout count is 74, and there are still 11 informal arguments, with the informal holdout count being 63.

Later, on [https://discord.com/channels/960643023006490684/1259770421046411285/1471227102844944510 11 Feb 2026], Andrew Ducharme found another machine nonhalting, again verified by @mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471228798505582602 the same day]. @mxdys also [https://discord.com/channels/960643023006490684/1259770421046411285/1471229409829847111 announced another TM as a translated cycler], thus reducing the holdout count to 72, with 61 informal holdouts.

On [https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 11 March 2026], Peacemaker II solved a permutation of one of the TMs Andrew solved by tweaking some of the decider parameters. This result was verified by @mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1481326301209165877 the same day,] reducing the holdout count to 71, with 60 informal holdouts.

== Holdouts ==
This section is based on the list of 83 holdouts published by @mxdys, and includes further progress as of 16 March 2026. There are 69 holdouts, or 60 when considering informal proofs.

=== Cryptids ===

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}. Hydra
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}. Bonus Cryptid

=== Unsolved ===

* {{TM|1RB2RA3LA4LA2RB_2LA3RA---0RA1LA|undecided}}. Chaotic via long. analysis - [https://discord.com/channels/960643023006490684/1259770421046411285/1436149296004071615 Notes by mxdys]
* {{TM|1RB2RA3LA4LA2RB_2LA3RB---0RA1LA|undecided}}. Chaotic via long. analysis
* {{TM|1RB2LA0RB4LB0LA_1LA3LA1RA4RA---|undecided}}. [https://discord.com/channels/960643023006490684/1471178503235043493/1471206925096980664 Does not halt in 1e13 steps.]
* {{TM|1RB---3RA2LA2RB_2LB3LA4LB4RA0RA|undecided}}. [https://discord.com/channels/960643023006490684/1471178503235043493/1471206925096980664 Does not halt in 1.25e13]
* {{TM|1RB3RB1LA2LA3RA_1LB2RA4RB0LA---|undecided}}. [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 Does not halt in 2e13 steps.]
* {{TM|1RB2RA4LA1RB4RB_1LB2LA3RA---0LB|undecided}}. [https://discord.com/channels/960643023006490684/1471178503235043493/1471206925096980664 Does not halt in 5e13 steps.]
* {{TM|1RB---4LB0LA4RA_2LB2LA3RA4LB0RB|undecided}}.
* {{TM|1RB4RA1LA4RB2LA_2LB3LA1RB2RA---|undecided}}.
* {{TM|1RB2RB3LA4LA1LA_2LB3RA---4RA1RB|undecided}}.
* {{TM|1RB3RB3LA4LA2RB_2LB3RA---1RA1LA|undecided}}.
* {{TM|1RB4RB4RA1LA3LA_1LB2LA3RB2RB---|undecided}}.
* {{TM|1RB3LA1LA2RB2RA_2LA4RA3LB1RA---|undecided}}.
* {{TM|1RB3RB---4RA2RA_2LA2RA3LB4LB1LB|undecided}}.
* {{TM|1RB---3LB4RB0LA_2LB3LA3RB4RA0RA|undecided}}.
* {{TM|1RB2LA0RB1LA3LB_1LA3LB1RA4RA---|undecided}}. Shift overflow mixed-digits counter - [https://discord.com/channels/960643023006490684/1440877223744770259/1440877223744770259 Analysis by hipparcos]
* {{TM|1RB2LA0RB4LB1RA_1LA3RA1RA---0LA|undecided}}. Shift overflow mixed-digits counter - [https://discord.com/channels/960643023006490684/1436181033992327333/1436181033992327333 Analysis by hipparcos] + [https://discord.com/channels/960643023006490684/1259770421046411285/1436151075450130443 mxdys's notes]
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}}. Potential Cryptid - [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}. Potential Cryptid
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}. Potential Cryptid
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}. Potential Cryptid - [https://discord.com/channels/960643023006490684/1354037062830919690/1354037062830919690 Shift overflow counter]
* {{TM|1RB2RA3LB---2LB_2LA0LA4RB0RB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1329808777754706046 30% chance of beating current champion]
* {{TM|1RB3LA1LA2RB2LB_1LB2RA4RA0RB---|undecided}}. [https://discord.com/channels/960643023006490684/1395820706050080869/1395820706050080869 Block analysis by @dyuan by "impurity score"]
* {{TM|1RB---4LB1RA4RA_2LB2LA3RA4LB0RB|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1378248683161653289 Analysis by Andrew Ducharme and @mxdys]
* {{TM|1RB---4RB2RB4LA_2LB3LA3LB0RA0RB|undecided}}. [https://discord.com/channels/960643023006490684/1353983911222312970/1353983911222312970 Bouncer + chaotic counter]
* {{TM|1RB2LA0RB0LB3LB_2LA4RB3RA0RA---|undecided}}. [https://discord.com/channels/960643023006490684/1395872756268269668/1395872756268269668 Analysis by Peacemaker II]
* {{TM|1RB2RA3LB4LA---_2LA0RB1LA2RB0RA|undecided}}. [https://discord.com/channels/960643023006490684/1348878717870673981 Analysis by @dyuan01 and @Legion]
* {{TM|1RB2RA3LA---2LB_2LA4RA4RB0RB0LA|undecided}}. Spaghetti, [https://discord.com/channels/960643023006490684/1344221797020602398/1344221797020602398 analysis by @nerdyjoe], [https://discord.com/channels/960643023006490684/1471178503235043493/1471206925096980664 does not halt in 4e13.]
* {{TM|1RB3LA3LB0RB0LA_2LA4RB1LB1RA---|undecided}}. Permutation of "Spaghetti TM", [https://discord.com/channels/960643023006490684/1344221797020602398 analysis by nerdyjoe]
* {{TM|1RB2RA3LA4LA2RB_2LA---3LB1RA3RA|undecided}}. [https://discord.com/channels/960643023006490684/1353983911222312970/1355112650690003028 Bouncer + chaotic counter]
* {{TM|1RB3LA3LA0RB2LB_2LA4LA4RA2RA---|undecided}}. [https://discord.com/channels/960643023006490684/1376383949575557161 Analysis by @mxdys]
* {{TM|1RB2LB3LA0RA1LB_2LA4RA3RB3LA---|undecided}}. [https://discord.com/channels/960643023006490684/1344221797020602398/1344221797020602398 Analysis by @nerdyjoe]
* {{TM|1RB3LB0RB---2LB_2LA3RA4RB2RB0LA|undecided}}. [https://discord.com/channels/960643023006490684/1376577295938097253 Analysis by @mxdys]
* {{TM|1RB3LB4LA0LB---_2LA0LA1RB0RA3RA|undecided}}. [https://discord.com/channels/960643023006490684/1395872756268269668/1395872756268269668 Analysis by Peacemaker II]
* {{TM|1RB3RB1LB2RA---_2LA2RB1LA4LB0RA|undecided}}. [https://discord.com/channels/960643023006490684/1397318518961082398 Analysis by Legion and @dyuan]
* {{TM|1RB2LA0RB---4LA_1LA3LA1RA4RA1LB|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1329629220795715674 Analysis by @racheline]
* {{TM|1RB4LA1RA1RB1LA_2LB3LA---4RA2RB|undecided}}. [https://discord.com/channels/960643023006490684/1359561443929886760 Basic long. analysis by @dyuan]
* {{TM|1RB3RA3RB4LA1LA_1LB2LA1LA---1RB|undecided}}. [https://discord.com/channels/960643023006490684/1349602897024782358 Long. analysis by @dyuan suggests chaotic, potentially halt]
* {{TM|1RB2LA4LA1RA1LA_2LB3RB4RB---2RA|undecided}}. [https://discord.com/channels/960643023006490684/1084047886494470185/1255570421437169805 Long. analysis rules by @Legion, ran to cell 155 without halting]
* {{TM|1RB2RB4LA2RA1LA_2LA4RA3LA---3RA|undecided}}. Chaotic via long. analysis. [https://discord.com/channels/960643023006490684/1353983911222312970/1353987502062702622 Probviously nonhalting]
* {{TM|1RB2LA4RA1LA3LA_0LA2RB3RB2LB---|undecided}}. 1D CA-like. [https://discord.com/channels/960643023006490684/1354107790330691655 Analysis by @dyuan and @mxdys]
* {{TM|1RB2LA4RA1LA3LA_0LA3RB3LB2RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA1LA4RA2LA_0LA3RB3LB2RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA3LA4RA1LA_0LA3LB3RB1RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA3LB4LB---_0LA4LB3RA4LA0RB|undecided}}. Fractal?
14 grandchildren of {{TM|1RB2LA0RB1LB_1LA3RA1RA---|undecided}}

* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4RB0LB|undecided}}.

and the family 1RB2LA0RB1LB---_1LA3RA1RA4LB---. See [https://discord.com/channels/960643023006490684/1336734852308799579 this thread] for more details.

* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2RB|undecided}}. Simulated for <math>~9*10^{1167}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2LB|undecided}}. Simulated for <math>~1.3*10^{1094}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1RB|undecided}}. Simulated for <math>~9.8*10^{1226}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1LB|undecided}}. Simulated for <math>~3*10^{1140}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0LB|undecided}}. Simulated for <math>~2.6*10^{889}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0RB|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB3RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2LA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1LA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0LA|undecided}}.

=== Solved with moderate rigor ===

* {{TM|1RB1RB3LA4LA2RA_2LB3RA---3RA4RB|undecided}}. [[Beaver Math Olympiad#3. 1RB0RB3LA4LA2RA 2LB3RA---3RA4RB (bbch) and 1RB1RB3LA4LA2RA 2LB3RA---3RA4RB (bbch)|BMO problem 3]] by @dyuan
* {{TM|1RB0RB3LA4LA2RA_2LB3RA---3RA4RB|undecided}}. BMO problem 3 by @dyuan
* {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|undecided}}. [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Longitudinal analysis by @Legion implies halting]
* {{TM|1RB3LA4LA1LA2RA_2LA4RB---0RA0LA|undecided}}. [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Longitudinal analysis by @Legion implies halting]
* {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}}. [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Longitudinal analysis by @Legion implies halting]
* {{TM|1RB2LB---4LB0RB_1LA3RB4RB4RA1LB|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1329663999700111471 Nonhalting argument by @racheline]
* {{TM|1RB3RA2LB1LB1RB_2LA2RA4LA1LA---|undecided}}. [[Dekaheptoid]], - [https://discord.com/channels/960643023006490684/1259770421046411285/1267650177389432913 Unverified nonhalting proof by @dyuan]
* {{TM|1RB3RB1LB---2RB_2LA1RA4LB2LA2RA|undecided}}. [[Dekaheptoid]], - [https://discord.com/channels/960643023006490684/1259770421046411285/1267650177389432913 Unverified nonhalting proof by @dyuan]
* {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB|undecided}}. BMO problem 3 variant - [https://discord.com/channels/960643023006490684/1259770421046411285/1415337575543214132 Nonhalting argument by @dyuan]

=== Formally proven ===

* {{TM|1RB2RA3LA4LA2RB_2LA0RA---0RA1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1355828077023854752 Rocq-decided by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1355714495326060706 Longitudinal analysis by Legion implies nonhalting]
* {{TM|1RB2RA3LA4RB---_2LA3RB3RA1LB3LB|halt}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 Rocq-decided by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1373347187836194898 Halting argument by @dyuan]
* {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 Rocq-decided by @mxdys]. Current champion
* {{TM|1RB3RA4LB2RA2RB_2LA---3LA0LB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 Rocq-decided by @mxdys.]
* {{TM|1RB3RB---0RA2RB_2LA4RA3LB1LB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 Rocq-decided by @mxdys.]
* {{TM|1RB0LB2LA4LB3LA_2LA---3RA4RB2RB|undecided}}. [https://discord.com/channels/960643023006490684/1375251026411786310/1375556785603084469 Rocq-decided by @mxdys]
* {{TM|1RB2LA0LB1LA2RA_0LA3RA1RA4LB---|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1428501877947109437 Nonhalting argument by Peacemaker II]. [https://discord.com/channels/960643023006490684/1259770421046411285/1483043657778069564 Confirmed in Rocq by @mxdys].
* {{TM|1RB2LA1RA---1LA_1LA4RB3LB0RB2RB|undecided}}. [https://discord.com/channels/960643023006490684/1375251026411786310/1375556785603084469 Rocq-decided by @mxdys]
* {{TM|1RB2LA3LA4RA0LA_1LA3RB1RB1LB---|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379521528869421137 Rocq-decided by @mxdys]
* {{TM|1RB2LA0RB1LB0LB_1LA3RA1RA4RA---|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1466208979511414885 Non-halting found by Andrew Ducharme]. [https://discord.com/channels/960643023006490684/1259770421046411285/1466331107279769736 Confirmed in Rocq by @mxdys]. Grandchild of {{TM|1RB2LA0RB1LB_1LA3RA1RA---|undecided}}.
* {{TM|1RB2RB---0LB3LA_2LA2LB3RB4RB1LB|undecided}}. Chaotic via long. analysis. [https://discord.com/channels/960643023006490684/1378560417235734558 Analysis of permutation by mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1471227102844944510 Non-halting found by Andrew Ducharme]. [https://discord.com/channels/960643023006490684/1259770421046411285/1471228798505582602 Confirmed in Rocq by @mxdys.]
* {{TM|1RB3LA1RA4LA2RA_2LA---1LA0RA3RB|undecided}}. Chaotic via long. analysis. [https://discord.com/channels/960643023006490684/1378560417235734558 More analysis by mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1472647706835943596 High-level behaviour by Peacemaker II.] [https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 Non-halting found by Peacemaker II.] [https://discord.com/channels/960643023006490684/1259770421046411285/1481326301209165877 Confirmed in Rocq by @mxdys.]
* {{TM|1RB3LA1LA4LA2RA_2LB2RA---0RA0RB|undecided}}. - [https://discord.com/channels/960643023006490684/1259770421046411285/1436151969986379868 Notes by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1471229409829847111 Translated cycler via @mxdys.]
* {{TM|1RB4LA1LB2LA0RB_2LB3RB4LA---1RA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1290449717536489622 Nonhalting argument by @dyuan]. [https://discord.com/channels/960643023006490684/1259770421046411285/1483043448855461989 Confirmed in Rocq by @mxdys].

[[Category:BB Domains]][[Category:BB(2,5)]]

BB(3,3)

2026-03-15T21:10:36Z

ADucharme: Updated Certified Progress

The 3-state, 3-symbol Busy Beaver problem, '''BB(3,3)''', is unsolved. With the discovery of the [[Cryptids|Cryptid]] machine [[Bigfoot]] in October 2023, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(3,3) and thus [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html BB(3,3) is Hard]. Of the remaining six with unknown behavior, there are two pairs of holdouts with equivalent behavior, so, in effect, there are four holdouts remaining.

The current BB(3,3) [[Champions#3-Symbol TMs|champion]] {{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}} was discovered by [[User:tjligocki|Terry]] and [[User:sligocki|Shawn Ligocki]] in November 2007, proving the lower bounds:

<math display="block">\begin{array}{lcrl}
S(3, 3) & \ge & 119{,}112{,}334{,}170{,}342{,}541 & > 10^{17} \\
\Sigma(3, 3) & \ge & 374{,}676{,}383 & > 10^8 \\
\end{array}</math>

It is believed that the [[probviously]] halting [[Cryptid]] {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}} beats this record, but it has not been proven.

== Cryptids ==
Known Cryptids:
* {{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}, known as [[Bigfoot]]
Potential Cryptids:

* {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}}, probable champion

== Top Halters ==
The current top 20 BB(3,3) halters (known by [[User:sligocki|Shawn Ligocki]]) are:
<pre>
Standard format Status S Σ
0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC Halt 119112334170342541 374676383
1RB2LA1LC_0LA2RB1LB_1RZ1RA1RC Halt 119112334170342540 374676383
1RB2RC1LA_2LA1RB1RZ_2RB2RA1LC Halt 4345166620336565 95524079
1RB1LA2LC_2LA2RB1RB_1RZ0LB0RC Halt 452196003014837 21264944
1RB1RZ2LC_1LC2RB1LB_1LA2RC2LA Halt 4144465135614 2950149
1RB2LA1RA_1RC2RB0RC_1LA1RZ1LA Halt 987522842126 1525688
1RB1RZ2RB_1LC0LB1RA_1RA2LC1RC Halt 4939345068 107900
1RB2LA1RA_1LB1LA2RC_1RZ1LC2RB Halt 1808669066 43925
1RB2LA1RA_1LC1LA2RC_1RZ1LA2RB Halt 1808669046 43925
1RB2LA1RA_1LB1LA2RC_1RZ1LA2RB Halt 1808669046 43925
1RB2LA1RA_1LC2RB1RC_1RZ1LA1LB Halt 1093389035 34151
1RB1LB2LA_1LA1RC1RZ_0LA2RC1LC Halt 544884219 32213
1RB2RA2RC_1LC1RZ1LA_1RA2LB1LC Halt 310341163 36089
1RB1RZ2LC_1LC2RB1LB_1LA0RB2LA Halt 92649163 13949
1RB2LA1LA_2LA1RC2RB_1RZ0LC0RA Halt 51525774 7205
1RB2RA1LA_2LA2LB2RC_1RZ2RB1RB Halt 47287015 12290
1RB2RA1LA_2LC0RC1RB_1RZ2LA1RB Halt 29403894 5600
1RB1LA2LA_1LB1RC1RZ_1LA2RC1LC Halt 15725661 4098
1RB1LA2LA_2RC1RC1RZ_1LA2RC1LC Halt 15725659 4098
1RB1LA1RZ_0LC2RB1LB_1RA1LC2LC Halt 15725629 4096
</pre>
Numbers listed are step count and sigma score for each TM. For a longer list of halting TMs see https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/3x3. For historical perspective see Pascal Michel's [https://bbchallenge.org/~pascal.michel/ha#tm33 '''Historical survey of Busy Beavers'''].

== Certified Progress ==
On 8 June 2023, Shawn Ligocki released a list of [https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 2417 BB(3,3) holdouts]. The next day, @iijil released their list of [https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 2380 holdouts]. The intersection of these two lists resulted in [https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 925 holdouts]. This list of 925 holdouts served as the progenitor list for further effort in BB(3,3), notably by Justin Blanchard. The ID system used for the BB(3,3) machines below are based on this list.

By 14 July 2024, this effort spearheaded by Justin Blanchard resulted in [https://discord.com/channels/960643023006490684/1259770474897080380/1262265485639286876 21 holdouts]. This list is accompanied by a file containing the 904 machines solved by Justin Blanchard and various other contributors over this time period.

On [https://discord.com/channels/960643023006490684/1259770474897080380/1325592156257124443 5 Jan 2025], @tjligocki finished an independent enumeration and filtering of the BB(3,3) machines using the established Ligocki filters. He also computed the number of steps and sigma scores for all found halting TMs. The pipeline used to solve all but 367 machines is described [https://docs.google.com/spreadsheets/d/1PU386wH1wcOpSAhU5BjJqkub7YJv5cRQ/edit?usp=share_link&ouid=101978079885839753033&rtpof=true&sd=true here]. He completed another filtering pass on [https://discord.com/channels/960643023006490684/1259770474897080380/1352398469456859167 20 Mar 2025] ([https://docs.google.com/spreadsheets/d/1HhbgIJiVjVU_G4MP6sBSdAlixVKxCu5o/edit?usp=sharing&ouid=101978079885839753033&rtpof=true&sd=true using some non-Ligocki deciders]) that left only 76 holdouts.

On [https://discord.com/channels/960643023006490684/1259770474897080380/1344554618188730368 26 Feb 2025], @mxdys published a list of 19 holdouts that withstood state-of-the-art Rocq deciders. Some of these machines already had fairly rigorous or even full Rocq proofs for non-halting, which were integrated into a 12 TM Rocq holdout list published on [https://discord.com/channels/960643023006490684/1259770474897080380/1409402854292066335 24 Aug 2025.] Then, on 27 Aug 2025, @mxdys certified two more machines in Rocq, lowering the Rocq holdout list to 10 TMs. Afterwards, on [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 30 Aug 2025], @mxdys certified one more machine in Rocq, lowering the Rocq holdout list to 9 TMs. The three remaining machines with moderate-rigor arguments were believed to never halt due to [[Longitudinal Analysis|longitudinal analysis]] by @Legion. @mxdys formally showed these holdouts to be non-halting on [https://discord.com/channels/960643023006490684/1259770474897080380/1482680295357677651 15 March 2026], uniting the informal and formal holdout lists at 6 TMs.

== Holdouts ==
This section is based on @mxdys's 27 August 2025 [[Holdouts lists|holdouts list]] of [https://discord.com/channels/960643023006490684/1259770474897080380/1410308974275985428 10 TMs.]

=== Cryptids ===
* {{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}} (829), Bigfoot

=== Unsolved ===
* {{TM|1RB1LB2LC_1LA2RB1RB_---0LA2LA|undecided}} (397), similar to Wily Coyote
* {{TM|1RB0LB0RC_2LC2LA1RA_1RA1LC---|undecided}} (153, equivalent to 758), potential Cryptid
* {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}} (758, equivalent to 153)
* {{TM|1RB2LA1LA_2LA0RA2RC_---0LC2RA|undecided}} (531, equivalent to 532), Wily Coyote
* {{TM|1RB2LA1LA_2LA0RA2RC_---1RB2RA|undecided}} (532, equivalent to 531)

=== Solved with moderate rigor ===
''None''

=== Formally proven ===
On 30 Aug 2025, @mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 proved in Rocq] that {{TM|1RB2RB1LC_1LA2RB0RB_2LB---0LA|undecided}} (867) is non-halting. LegionMammal had previously provided some [https://discord.com/channels/960643023006490684/1084047886494470185/1256634331464601640 longitudinal analysis that implied non-halting].

On 15 March 2026, @mxdys [https://discord.com/channels/960643023006490684/1259770474897080380/1482680295357677651 formalised] into Rocq the following informal results:

*{{TM|1RB2LB0LC_2LA2RA1RB_---2LA1LC|undecided}} (650) ([https://cosearch.bbchallenge.org/contribution/32099ui7 cosearch]). [https://discord.com/channels/960643023006490684/1259770474897080380/1290044219481657476 Longitudinal analysis (with extra typo disclaimer) by @Legion implies non-halting]
* {{TM|1RB1LC1LC_1LA2RB0RB_2LB---0LA|undecided}} (412) ([https://cosearch.bbchallenge.org/contribution/f9gdw0mg cosearch]). [https://discord.com/channels/960643023006490684/1259770474897080380/1279055891559223399 Longitudinal analysis by @Legion implies non-halting]
* {{TM|1RB0RC---_2RC0LB1LB_2LC2RA2RB|undecided}} (279). [https://discord.com/channels/960643023006490684/1084047886494470185/1256729514076016733 Longitudinal analysis by @Legion implies non-halting]

== Interesting Rocq-Solved TMs ==
The following TMs have halting problems highly dependent on that of machine 816. While all TMs were solved individually, it was theoretically possible that someone solved machine 816 and solved up to five machines "for free." If 816 was non-halting, then 21, 92, 683, 817, and 818 were all non-halting. If 816 halted via transition C0, then 817 halted. And if 816 halted via transition C2, then 21, 92, 683 and 818 all halted. A compilation of the various analyses can be found [https://discord.com/channels/960643023006490684/1259770474897080380/1291694606534049793 here]
* {{TM|1RB---0LC_2LC2RC1LB_0RA2RB0LB|undecided}} (21, equivalent to 92 and 818) [https://discord.com/channels/960643023006490684/1084047886494470185/1256634331464601640 Longitudinal analysis by @Legion implies non-halting]
* {{TM|1RB---1RB_2LC2RC1LB_0RA2RB0LB|undecided}} (92, equivalent to 21 and 818) [https://discord.com/channels/960643023006490684/1084047886494470185/1213612098232262707 Equivalence claim to 21 by @dyuan1]
* {{TM|1RB2RA1LB_0LC0RA1LA_2LA0RB---|undecided}} (818, equivalent to 21 and 92)
* {{TM|1RB2LC---_0LA0RC1LC_1RB2RC1LB|undecided}} (683)
* {{TM|1RB2RA1LB_0LC0RA1LA_---2LA---|undecided}} (816) [https://discord.com/channels/960643023006490684/1259770474897080380/1278398402450817025 See discussion] of likely non-halting by @Rae and @Peacemaker on 28 August 2024
* {{TM|1RB2RA1LB_0LC0RA1LA_---2RB2LA|undecided}} (817)
These TMs were on Justin Blanchard's informal [[Holdouts lists|holdouts list]] of 22 TMs but were Rocq-decided individually [https://discord.com/channels/960643023006490684/1259770474897080380/1344554618188730368 by @mxdys] in their February 2025 release. Two other members of Justin Blanchard's list Rocq-decided by mxdys in February 2025 were {{TM|1RB2LB---_1RC2RB1LC_0LA0RB1LB|undecided}} (642) and {{TM|1RB2RB---_1LC2LB1RC_0RA0LB1RB|undecided}} (834). @-d independently generated a Rocq proof for 642 ([https://cosearch.bbchallenge.org/contribution/3wxkcx6f cosearch]), and @dyuan01 independently discovered non-halting arguments for [https://discord.com/channels/960643023006490684/1084047886494470185/1222694817771814922 642] and [https://discord.com/channels/960643023006490684/1084047886494470185/1222694817771814922 834], and noted their similarity.

Similarly, @-d independently wrote a [https://discord.com/channels/960643023006490684/1218877181321678928/1265131102289264690 Rocq proof] for {{TM|1RB2LA0LA_2LC---2RA_0RA2RC1LC|undecided}} (494) which was adapted into mxdys's pipeline in August of 2025. This release also adapted the proof to formally prove {{TM|1RB1LC---_0LC2RB1LB_2LA0RC1RC|undecided}} (400), which was known to be [https://discord.com/channels/960643023006490684/1084047886494470185/1215532512307052634 equivalent to 494.]

The August release also confirmed the non-halting status of {{TM|1RB2LA1RA_1LC1RC2RB_---1RA1LC|undecided}} (575), {{TM|1RB2LA1RA_1LC2RC2RB_---1RA1LC|undecided}} (585), and {{TM|1RB2LA1RA_2LC2RC2RB_---2LA1LC|undecided}} (588), which, as first shown by [https://discord.com/channels/960643023006490684/1084047886494470185/1224879612165881888 @dyuan01 and @Justin Blanchard,] infinitely enumerate the series <math>p_n = a 2^n - (-1)^n b - \lfloor n / 2 \rfloor - c</math> for machine-dependent constants a, b, and c.

On August 27, 2025, @mxdys proved in Rocq that {{TM|1RB2LA1LC_1LA2RB1RB_---2LB0LC|undecided}} (543) is non-halting, formalizing arguments for non-halting made by @dyuan for on [https://discord.com/channels/960643023006490684/1084047886494470185/1240854539519791216 16 May 2024] and [https://discord.com/channels/960643023006490684/1259770474897080380/1325646208483590177 5 January 2025]. On the same day, @mxdys confirmed in Rocq the non-halting status of {{TM|1RB2LA1LA_0LA0RC0LC_---2RA1RA|undecided}} (522), which @Justin Blanchard had already generated [https://discord.com/channels/960643023006490684/1259770474897080380/1262265485639286876 normal] and [https://discord.com/channels/960643023006490684/1259770474897080380/1409497663166087271 two-sided] FAR certificates for.
[[Category:BB Domains]][[Category:BB(3,3)]]

User:ADucharme

2026-03-13T05:34:36Z

ADucharme: /* BB(6) */ add decider BB(6) machines

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts since that enumeration. I've also tried my hand at the analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last ~1500 BB(6) holdouts, I solved 67 and counting. Partial credit for some of these machines goes to Peacemaker II, who identifies permutations of machines I solved in the holdout list. Because of the shared behavior between permutations, I can apply the decider which solved to the original TM I found to the permutations, and often solve permutations too.

Solved halting TMs (49) with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs (18) with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9
1RB0RB_1LC1RA_0LA1RD_1LA1LE_1LF1LD_---0LC FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 3 n 12
1RB0LD_1RC0RE_0LA0RC_1LA1LD_0RF1RA_---1RC FAR CPS_LRU maxT 10000000 LRUH 6 H 3 tH 0 n 9
1RB1LB_1LC1RE_0RD0LB_0LB1RA_1LA0RF_---0RC FAR CPS_LRU maxT 10000000 LRUH 7 H 3 tH 1 n 4
1RB0LD_0RC1RF_1RD0RA_1LE1RB_1LC0LE_1RC--- FAR CPS_LRU maxT 10000000 LRUH 7 H 4 tH 1 n 24
1RB0LA_0RC---_1RD1RE_1LA1LD_1RD0RF_0RC1RC FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB0LA_1RC1RA_0LD1LA_1LF1RE_0RD0RE_0LC--- FAR RWL_mod maxT 10000000 H 4 mod 1 n 8
1RB1RF_1LC1LB_---0LD_1RE0LD_0RA1RA_0LE0RE FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB1RD_0RC1RE_1LD0RE_1LB---_0RA1LF_0LE0LF FAR CPS_LRU maxT 1000000 LRUH 32 H 1 tH 29 n 12
1RB1RE_1LC0RF_1RE0LD_1LC0LB_1RA0RE_1RC--- FAR CPS_LRU maxT 1000000 LRUH 32 H 4 tH 20 n 24
1RB0RE_1LC1RA_0LA1LD_1RE1LC_0RF1RB_---0LC FAR CPS_LRU maxT 1000000 LRUH 17 H 4 tH 13 n 3

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Games ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain. These remain my best-ever entries in these particular Busy Beaver games.

==== BB(7) ====
{| class="wikitable sortable"
|Place
|TM
|Score
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|5
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG}}
|10 ↑↑ 403.84
|-
|9
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}}
|10 ↑↑ 243.88
|}

{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}} was a bit of co-discovery: Iijil first enumerated the TM and I first showed it was halting.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}

BB(2,5)

2026-03-11T17:26:38Z

ADucharme: /* Holdouts */ new TM solved+verified 3/11/26

The 2-state, 5-symbol Busy Beaver problem, '''BB(2,5)''', is unsolved. With the discovery of the [[Cryptids|Cryptid]] machine [[Hydra]] in April 2024, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(2,5) and thus [https://www.sligocki.com/2024/05/10/bb-2-5-is-hard.html BB(2,5) is Hard].

The current BB(2,5) [[Champions#5-Symbol TMs|champion]] {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}} was discovered by Daniel Yuan in June 2024, proving the lower bounds:

<math display="block">S(2,5) > \Sigma(2,5) > 10^{10^{10^{3\,314\,360}}} > 10 \uparrow\uparrow 4</math>

== Cryptids ==
Known Cryptids:

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA}}, known as [[Hydra]]
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}, known as the [[Bonus Cryptid]]
Potential Cryptids:

* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}. [https://discord.com/channels/960643023006490684/1354037062830919690/1354037062830919690 Shift overflow counter]
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}.
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}.
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}}. [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]
* {{TM|1RB2RA3LB---2LB_2LA0LA4RB0RB1LA}}. Probviously halting. 1/8 chance of beating champ.

==Top Halters==
The 20 longest running known halting BB(2,5) TMs are:
{| class="wikitable"
|+
!Standard format
!(approximate) runtime
!Discoverer
|-
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|<math>10 \uparrow\uparrow 4.8142742</math>
|Daniel Yuan
|-
|{{TM|1RB2LB4LB3LA1RZ_1LA3RA3LB0LB0RA|halt}}
|<math>>10^{38\,033}</math>
|Pavel Kropitz
|-
|{{TM|1RB2LA1RA2LB2LA_0LA2RB3RB4RA1RZ|halt}}
|<math>>1.9 \times 10^{704}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2RA3LA4RB---_2LA3RB3RA1LB3LB|halt}}
|<math>>8.3 \times 10^{466}</math> (lower bound given by score)
|Daniel Yuan
|-
|{{TM|1RB2LA4RA2LB2LA_0LA2RB3RB1RA1RZ|halt}}
|<math>>1.6 \times 10^{211}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2LA4RA2LB2LA_0LA2RB3RB4RA1RZ|halt}}
|<math>>1.6 \times 10^{211}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2LA4RA1LB2LA_0LA2RB3RB2RA1RZ|halt}}
|<math>>5.2 \times 10^{61}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB0RB4RA2LB2LA_2LA1LB3RB4RA1RZ|halt}}
|<math>>7 \times 10^{21}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB1RZ4LA4LB2RA_2LB2RB3RB2RA0RB|halt}}
|<math>>9 \times 10^{16}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB3LA1LA0LB1RA_2LA4LB4LA1RA1RZ|halt}}
|<math>>3.77 \times 10^{16}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2RA1LA3LA2RA_2LA3RB4LA1LB1RZ|halt}}
|<math>>9 \times 10^{15}</math>
|Terry and Shawn Ligocki
|-
|{{TM|1RB2RA1LA1LB3LB_2LA3RB1RZ4RA1LA|halt}}
|417,310,842,648,366
|Terry and Shawn Ligocki
|-
|{{TM|1RB3LA1LA4LA1RA_2LB2RA1RZ0RA0RB|halt}}
|26,375,397,569,930
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB3LB4LB4LA2RA_2LA1RZ3RB4RA3RB|halt}}
|14,103,258,269,249
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB3RA4LB2RA3LA_2LA1RZ4RB4RB2LB|halt}}
|3,793,261,759,791
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB3RA1LA1LB3LB_2LA4LB3RA2RB1RZ|halt}}
|924,180,005,181
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB3LB1RZ1LA1LA_2LA3RB4LB4LB3RA|halt}}
|912,594,733,606
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB2RB3LA2RA3RA_2LB2LA3LA4RB1RZ|halt}}
|469,121,946,086
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB3RB3RB1LA3LB_2LA3RA4LB2RA1RZ|halt}}
|233,431,192,481
|Grégory Lafitte and Christophe Papazian
|-
|{{TM|1RB3LA1LB1RA3RA_2LB3LA3RA4RB1RZ|halt}}
|8,619,024,596
|Grégory Lafitte and Christophe Papazian
|}

== Certified progress ==
In April 2024, Shawn Ligocki publicly released a list of 23,411 undecided BB(2,5) machines. Justin Blanchard then made substantial progress over the course of the next month, reducing the list to 499 [[Holdouts lists|holdouts]] by late May 2024. In June 2024, @mxdys cut down the list to 273 using halting and inductive deciders, and again to 217 using [[Closed Tape Language|CTL]]. In February 2025, @mxdys ran a decider pipeline in Rocq that resulted in only 173 holdouts. Since then, additional machines have been proven in Rocq using both deciders and individual proofs.

On [https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 29 Mar 2025], @mxdys published a list of 83 holdouts that withstood state-of-the-art Rocq deciders.

Over the course of 5 months, @mxdys added 8 machines to Rocq[https://discord.com/channels/960643023006490684/1259770421046411285/1355799763437752521 1][https://discord.com/channels/960643023006490684/1259770421046411285/1355828077023854752 2][https://discord.com/channels/960643023006490684/1259770421046411285/1379521528869421137 3][https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 4][https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 5], lowering the certified holdout count to 75. There are 11 informal arguments, lowering the informal holdout count to 64.

Then, on [https://discord.com/channels/960643023006490684/1259770421046411285/1466208979511414885 29 Jan 2026], Andrew Ducharme found a machine nonhalting. This was verified by @mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1466331107279769736 the same day]. Hence the certified holdout count is 74, and there are still 11 informal arguments, with the informal holdout count being 63.

Later, on [https://discord.com/channels/960643023006490684/1259770421046411285/1471227102844944510 11 Feb 2026], Andrew Ducharme found another machine nonhalting, again verified by @mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471228798505582602 the same day]. @mxdys also [https://discord.com/channels/960643023006490684/1259770421046411285/1471229409829847111 announced another TM as a translated cycler], thus reducing the holdout count to 72, with 61 informal holdouts.

On [https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 11 March 2026], Peacemaker II solved a permutation of one of the TMs Andrew solved by tweaking some of the decider parameters. This result was verified by @mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1481326301209165877 the same day,] reducing the holdout count to 71, with 60 informal holdouts.

== Holdouts ==
This section is based on the list of 83 holdouts published by @mxdys, and includes further progress as of 11 March 2026. There are 71 holdouts, or 60 when considering informal proofs.

=== Cryptids ===

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA|undecided}}. Hydra
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB|undecided}}. Bonus Cryptid

=== Unsolved ===

* {{TM|1RB2RA3LA4LA2RB_2LA3RA---0RA1LA|undecided}}. Chaotic via long. analysis - [https://discord.com/channels/960643023006490684/1259770421046411285/1436149296004071615 Notes by mxdys]
* {{TM|1RB2RA3LA4LA2RB_2LA3RB---0RA1LA|undecided}}. Chaotic via long. analysis
* {{TM|1RB2LA0RB4LB0LA_1LA3LA1RA4RA---|undecided}}. [https://discord.com/channels/960643023006490684/1471178503235043493/1471206925096980664 Does not halt in 1e13 steps.]
* {{TM|1RB---3RA2LA2RB_2LB3LA4LB4RA0RA|undecided}}. [https://discord.com/channels/960643023006490684/1471178503235043493/1471206925096980664 Does not halt in 1.25e13]
* {{TM|1RB3RB1LA2LA3RA_1LB2RA4RB0LA---|undecided}}. [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 Does not halt in 2e13 steps.]
* {{TM|1RB2RA4LA1RB4RB_1LB2LA3RA---0LB|undecided}}. [https://discord.com/channels/960643023006490684/1471178503235043493/1471206925096980664 Does not halt in 5e13 steps.]
* {{TM|1RB---4LB0LA4RA_2LB2LA3RA4LB0RB|undecided}}.
* {{TM|1RB4RA1LA4RB2LA_2LB3LA1RB2RA---|undecided}}.
* {{TM|1RB2RB3LA4LA1LA_2LB3RA---4RA1RB|undecided}}.
* {{TM|1RB3RB3LA4LA2RB_2LB3RA---1RA1LA|undecided}}.
* {{TM|1RB4RB4RA1LA3LA_1LB2LA3RB2RB---|undecided}}.
* {{TM|1RB3LA1LA2RB2RA_2LA4RA3LB1RA---|undecided}}.
* {{TM|1RB3RB---4RA2RA_2LA2RA3LB4LB1LB|undecided}}.
* {{TM|1RB---3LB4RB0LA_2LB3LA3RB4RA0RA|undecided}}.
* {{TM|1RB2LA0RB1LA3LB_1LA3LB1RA4RA---|undecided}}. Shift overflow mixed-digits counter - [https://discord.com/channels/960643023006490684/1440877223744770259/1440877223744770259 Analysis by hipparcos]
* {{TM|1RB2LA0RB4LB1RA_1LA3RA1RA---0LA|undecided}}. Shift overflow mixed-digits counter - [https://discord.com/channels/960643023006490684/1436181033992327333/1436181033992327333 Analysis by hipparcos] + [https://discord.com/channels/960643023006490684/1259770421046411285/1436151075450130443 mxdys's notes]
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}}. Potential Cryptid - [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}. Potential Cryptid
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}. Potential Cryptid
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}. Potential Cryptid - [https://discord.com/channels/960643023006490684/1354037062830919690/1354037062830919690 Shift overflow counter]
* {{TM|1RB2RA3LB---2LB_2LA0LA4RB0RB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1329808777754706046 30% chance of beating current champion]
* {{TM|1RB3LA1LA2RB2LB_1LB2RA4RA0RB---|undecided}}. [https://discord.com/channels/960643023006490684/1395820706050080869/1395820706050080869 Block analysis by @dyuan by "impurity score"]
* {{TM|1RB---4LB1RA4RA_2LB2LA3RA4LB0RB|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1378248683161653289 Analysis by Andrew Ducharme and @mxdys]
* {{TM|1RB---4RB2RB4LA_2LB3LA3LB0RA0RB|undecided}}. [https://discord.com/channels/960643023006490684/1353983911222312970/1353983911222312970 Bouncer + chaotic counter]
* {{TM|1RB2LA0RB0LB3LB_2LA4RB3RA0RA---|undecided}}. [https://discord.com/channels/960643023006490684/1395872756268269668/1395872756268269668 Analysis by Peacemaker II]
* {{TM|1RB2RA3LB4LA---_2LA0RB1LA2RB0RA|undecided}}. [https://discord.com/channels/960643023006490684/1348878717870673981 Analysis by @dyuan01 and @Legion]
* {{TM|1RB2RA3LA---2LB_2LA4RA4RB0RB0LA|undecided}}. Spaghetti, [https://discord.com/channels/960643023006490684/1344221797020602398/1344221797020602398 analysis by @nerdyjoe], [https://discord.com/channels/960643023006490684/1471178503235043493/1471206925096980664 does not halt in 4e13.]
* {{TM|1RB3LA3LB0RB0LA_2LA4RB1LB1RA---|undecided}}. Permutation of "Spaghetti TM", [https://discord.com/channels/960643023006490684/1344221797020602398 analysis by nerdyjoe]
* {{TM|1RB2RA3LA4LA2RB_2LA---3LB1RA3RA|undecided}}. [https://discord.com/channels/960643023006490684/1353983911222312970/1355112650690003028 Bouncer + chaotic counter]
* {{TM|1RB3LA3LA0RB2LB_2LA4LA4RA2RA---|undecided}}. [https://discord.com/channels/960643023006490684/1376383949575557161 Analysis by @mxdys]
* {{TM|1RB2LB3LA0RA1LB_2LA4RA3RB3LA---|undecided}}. [https://discord.com/channels/960643023006490684/1344221797020602398/1344221797020602398 Analysis by @nerdyjoe]
* {{TM|1RB3LB0RB---2LB_2LA3RA4RB2RB0LA|undecided}}. [https://discord.com/channels/960643023006490684/1376577295938097253 Analysis by @mxdys]
* {{TM|1RB3LB4LA0LB---_2LA0LA1RB0RA3RA|undecided}}. [https://discord.com/channels/960643023006490684/1395872756268269668/1395872756268269668 Analysis by Peacemaker II]
* {{TM|1RB3RB1LB2RA---_2LA2RB1LA4LB0RA|undecided}}. [https://discord.com/channels/960643023006490684/1397318518961082398 Analysis by Legion and @dyuan]
* {{TM|1RB2LA0RB---4LA_1LA3LA1RA4RA1LB|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1329629220795715674 Analysis by @racheline]
* {{TM|1RB4LA1RA1RB1LA_2LB3LA---4RA2RB|undecided}}. [https://discord.com/channels/960643023006490684/1359561443929886760 Basic long. analysis by @dyuan]
* {{TM|1RB3RA3RB4LA1LA_1LB2LA1LA---1RB|undecided}}. [https://discord.com/channels/960643023006490684/1349602897024782358 Long. analysis by @dyuan suggests chaotic, potentially halt]
* {{TM|1RB2LA4LA1RA1LA_2LB3RB4RB---2RA|undecided}}. [https://discord.com/channels/960643023006490684/1084047886494470185/1255570421437169805 Long. analysis rules by @Legion, ran to cell 155 without halting]
* {{TM|1RB2RB4LA2RA1LA_2LA4RA3LA---3RA|undecided}}. Chaotic via long. analysis. [https://discord.com/channels/960643023006490684/1353983911222312970/1353987502062702622 Probviously nonhalting]
* {{TM|1RB2LA4RA1LA3LA_0LA2RB3RB2LB---|undecided}}. 1D CA-like. [https://discord.com/channels/960643023006490684/1354107790330691655 Analysis by @dyuan and @mxdys]
* {{TM|1RB2LA4RA1LA3LA_0LA3RB3LB2RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA1LA4RA2LA_0LA3RB3LB2RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA3LA4RA1LA_0LA3LB3RB1RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA3LB4LB---_0LA4LB3RA4LA0RB|undecided}}. Fractal?
14 grandchildren of {{TM|1RB2LA0RB1LB_1LA3RA1RA---|undecided}}

* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4RB0LB|undecided}}.

and the family 1RB2LA0RB1LB---_1LA3RA1RA4LB---. See [https://discord.com/channels/960643023006490684/1336734852308799579 this thread] for more details.

* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2RB|undecided}}. Simulated for <math>~9*10^{1167}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2LB|undecided}}. Simulated for <math>~1.3*10^{1094}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1RB|undecided}}. Simulated for <math>~9.8*10^{1226}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1LB|undecided}}. Simulated for <math>~3*10^{1140}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0LB|undecided}}. Simulated for <math>~2.6*10^{889}</math> steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0RB|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB3RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2LA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1LA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0LA|undecided}}.

=== Solved with moderate rigor ===

* {{TM|1RB1RB3LA4LA2RA_2LB3RA---3RA4RB|undecided}}. [[Beaver Math Olympiad#3. 1RB0RB3LA4LA2RA 2LB3RA---3RA4RB (bbch) and 1RB1RB3LA4LA2RA 2LB3RA---3RA4RB (bbch)|BMO problem 3]] by @dyuan
* {{TM|1RB0RB3LA4LA2RA_2LB3RA---3RA4RB|undecided}}. BMO problem 3 by @dyuan
* {{TM|1RB2RA3LA4LA2RB_2LA---1LA1RA3RA|undecided}}. [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Longitudinal analysis by @Legion implies halting]
* {{TM|1RB3LA4LA1LA2RA_2LA4RB---0RA0LA|undecided}}. [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Longitudinal analysis by @Legion implies halting]
* {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|undecided}}. [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Longitudinal analysis by @Legion implies halting]
* {{TM|1RB2LB---4LB0RB_1LA3RB4RB4RA1LB|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1329663999700111471 Nonhalting argument by @racheline]
* {{TM|1RB2LA0LB1LA2RA_0LA3RA1RA4LB---|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1428501877947109437 Nonhalting argument by Peacemaker II]
* {{TM|1RB4LA1LB2LA0RB_2LB3RB4LA---1RA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1290449717536489622 Nonhalting argument by @dyuan]
* {{TM|1RB3RA2LB1LB1RB_2LA2RA4LA1LA---|undecided}}. [[Dekaheptoid]], - [https://discord.com/channels/960643023006490684/1259770421046411285/1267650177389432913 Unverified nonhalting proof by @dyuan]
* {{TM|1RB3RB1LB---2RB_2LA1RA4LB2LA2RA|undecided}}. [[Dekaheptoid]], - [https://discord.com/channels/960643023006490684/1259770421046411285/1267650177389432913 Unverified nonhalting proof by @dyuan]
* {{TM|1RB0RA3LA4LA2RA_2LB3LA---4RA3RB|undecided}}. BMO problem 3 variant - [https://discord.com/channels/960643023006490684/1259770421046411285/1415337575543214132 Nonhalting argument by @dyuan]

=== Formally proven ===

* {{TM|1RB2RA3LA4LA2RB_2LA0RA---0RA1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1355828077023854752 Rocq-decided by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1355714495326060706 Longitudinal analysis by Legion implies nonhalting]
* {{TM|1RB2RA3LA4RB---_2LA3RB3RA1LB3LB|halt}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 Rocq-decided by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1373347187836194898 Halting argument by @dyuan]
* {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 Rocq-decided by @mxdys]. Current champion
* {{TM|1RB3RA4LB2RA2RB_2LA---3LA0LB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 Rocq-decided by @mxdys.]
* {{TM|1RB3RB---0RA2RB_2LA4RA3LB1LB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 Rocq-decided by @mxdys.]
* {{TM|1RB0LB2LA4LB3LA_2LA---3RA4RB2RB|undecided}}. [https://discord.com/channels/960643023006490684/1375251026411786310/1375556785603084469 Rocq-decided by @mxdys]
* {{TM|1RB2LA1RA---1LA_1LA4RB3LB0RB2RB|undecided}}. [https://discord.com/channels/960643023006490684/1375251026411786310/1375556785603084469 Rocq-decided by @mxdys]
* {{TM|1RB2LA3LA4RA0LA_1LA3RB1RB1LB---|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379521528869421137 Rocq-decided by @mxdys]
* {{TM|1RB2LA0RB1LB0LB_1LA3RA1RA4RA---|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1466208979511414885 Non-halting found by Andrew Ducharme]. [https://discord.com/channels/960643023006490684/1259770421046411285/1466331107279769736 Confirmed in Rocq by @mxdys]. Grandchild of {{TM|1RB2LA0RB1LB_1LA3RA1RA---|undecided}}.
* {{TM|1RB2RB---0LB3LA_2LA2LB3RB4RB1LB|undecided}}. Chaotic via long. analysis. [https://discord.com/channels/960643023006490684/1378560417235734558 Analysis of permutation by mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1471227102844944510 Non-halting found by Andrew Ducharme]. [https://discord.com/channels/960643023006490684/1259770421046411285/1471228798505582602 Confirmed in Rocq by @mxdys.]
* {{TM|1RB3LA1RA4LA2RA_2LA---1LA0RA3RB|undecided}}. Chaotic via long. analysis. [https://discord.com/channels/960643023006490684/1378560417235734558 More analysis by mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1472647706835943596 High-level behaviour by Peacemaker II.] [https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 Non-halting found by Peacemaker II.] [https://discord.com/channels/960643023006490684/1259770421046411285/1481326301209165877 Confirmed in Rocq by @mxdys.]
* {{TM|1RB3LA1LA4LA2RA_2LB2RA---0RA0RB|undecided}}. - [https://discord.com/channels/960643023006490684/1259770421046411285/1436151969986379868 Notes by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1471229409829847111 Translated cycler via @mxdys.]

[[Category:BB Domains]][[Category:BB(2,5)]]

Terminating Turmite

2026-03-08T07:24:07Z

ADucharme: /* History */ add reference to 2D work by Hutton et al

A '''Terminating Turmite''' or '''Relative Movement Turing Machine''' is a 1 dimentional [[Turing machine]] which uses relative directions instead of absolute ones. So instead of moving (L)eft or (R)ight, it (P)roceeds forward (for one step in the same direction as last move) or (T)urns-around (move one direction in the opposite direction). TT(n,k) is the maximum steps of all halting n-state, k-symbol Terminating Turmites when started on a blank tape.

== History ==
2D [[wikipedia:Turmite|Turmites]], also called '''turNing machines''', have been historically studied by Chris Langton in 1986 ([[wikipedia:Langton's_ant|Langton's ants]]), Allen Brady in 1988 (TurNing machines), and Greg Turk in 1989 (tur-mites). A [https://github.com/GollyGang/ruletablerepository/wiki/TwoDimensionalTuringMachines collaborative effort] led by Tim Hutton since 2011 has considered turmites on 2D square, triangular, and hexagonal grids. Until recently, it seems like much less investigation was put into 1D Turmites.

==Values==
{| class="wikitable"
!Domain
!Value
!Champion
|-
|TT(2)
|≥ 13
|<code>1TB---_1PA0PB</code>
|-
|TT(3)
|≥ 82
|<code>1PB0PA_1TA0PC_1PA---</code>
|-
|TT(4)
|≥ 48,186
|<code>1TB1PA_1PC0PA_1TA0PD_---1TA</code>
|-
|TT(2,3)
|≥ 223
|<code>1TB0PA2PA_2PA---1PA</code>
|-
|TT(3,3)
|≥ 45,153
|<code>1PB1PA1TA_2TB2PB2PC_---2PA1TC</code>
|-
|TT(2,4)
|> 3.467*1015
|<code>1TA2PB3TB---_3TA1PB1TA1PA</code>
|}

== See Also ==
* Google Sheet recording known values: https://docs.google.com/spreadsheets/d/18EXcLXM4Xb_qpKenV4oRGQQpCd45MJ4uawNWgmVvKTY/edit?gid=0#gid=0
[[category:Functions]]

TMBR: January 2026

2026-03-06T20:00:35Z

ADucharme: addition of 2x6 to summary. 35.9% reduction in the month is very notable imo!

{{TMBRnav|December 2025|February 2026}}

This is the first edition of TMBR in 2026. This month, we saw a rare occurrence: a [[BB(2,5)]] TM was proven to be nonhalting, by Andrew Ducharme. The BB(6), BB(2,6), and BB(7) holdout reductions were also notable. These were aided by a C++ implementation of FAR with a DFA generator released by mxdys on the 18th - see [https://discord.com/channels/960643023006490684/1239205785913790465/1462674863398326362 Discord].

== Champions ==

* nickdrozd discovered a new [[BLB|BLB(3,3)]] [[champion]] ({{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}) which blanks the tape after running for more than 1042,745 steps.
* creeperman7002 discovered <code>1TB1PA_1PC0PA_1TA0PD_---1TA</code>, a [[TT|TT(4,2)]] TM which runs for 48,186 steps and <code>1TA2PB3TB---_3TA1PB1TA1PA</code>, a [[TT|TT(2,4)]] TM which runs for more than 3.467*1015 steps.
* A new lower bound of <math>f_{\varepsilon_0 \omega^{\omega^{3}}}\left(4\right)</math> was computed for the [[Busy Beaver for lambda calculus#Champions|BBλ(91)]] champion.
* New champions were discovered for BBλ(61), BBλ(86), BBλ(90), BBλ(94), BBλ(96) and BBλ(100).

== Blog Posts ==

* 16 Jan 2026. Nick Drozd. [https://nickdrozd.github.io/2026/01/16/ai-lin-busy-beaver.html AI Reproduction of Lin's Busy Beaver Proof].
* 28 Jan 2026. John Tromp. [https://tromp.github.io/blog/2026/01/28/largest-number-revised The largest number representable in 64 bits - Revised].

== BB Adjacent ==

* [https://discord.com/channels/960643023006490684/1458010522967609425/1458010522967609425 Uniform-Action Busy Beavers] were introduced, and lower bounds have been given up to BBu(6) (with BBu(2)=6, BBu(3)=17 and BBu(4)=29)

== Methods ==
mxdys released C++ implementation of FAR with a DFA generator. See [https://discord.com/channels/960643023006490684/1239205785913790465/1462674863398326362 Discord].

== Theory ==
@ConePine [https://discord.com/channels/960643023006490684/960643023530762341/1458876275749032232 shared an idea] on how the value of BB(5) can be proved without enumerating Turing machines.

== Holdouts ==

{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!New Holdout Count
!Previous Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(2,5)]]
|74
|75
|1
|1.3%
|-
|[[BB(6)]]
|1314
|1326
|12
|0.9%
|-
|[[BB(7)]]
|19,303,801
|20,387,509
|1,083,708
|5.32%
|-
|[[BB(2,6)]]
|558,039
|870,085
|312,046
|35.86%
|}
* [[BB(6)]]: '''15''' solved machines. 1311 holdouts. 2 informally solved this month.
** Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire and aparker: see [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 spreadsheet]. Current count: '''205'''. (Total reduction: '''73''' (+1 later). 39 machines[https://discord.com/channels/960643023006490684/1456755170527543420/1456755170527543420 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1456757189787127971/1456757189787127971 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1456757980203585678/1456757980203585678 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1456758622762696815/1456758622762696815 <nowiki>[4]</nowiki>][https://discord.com/channels/960643023006490684/1456759118818971709/1456759118818971709 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1456761765688901724/1456761765688901724 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1456762231491657830/1456762231491657830 <nowiki>[7]</nowiki>][https://discord.com/channels/960643023006490684/1456310285462802587/1456760768472289321 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456932659740545217 <nowiki>[9]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456933097009320089 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456933714629099593 <nowiki>[11]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456934113943621653 <nowiki>[12]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457147385309565021 <nowiki>[13]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457409447084429594 <nowiki>[11 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457497643478683770 <nowiki>[4 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457644806984437760 <nowiki>[10 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457992404597739672 <nowiki>[11 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1458413343088840734 <nowiki>[17 more]</nowiki>][https://discord.com/channels/960643023006490684/1458063260959113279/1468676578455064742 <nowiki>[aparker's machine]</nowiki>] simulated out, plus 6 solved machines).
** Alistaire found a halting machine in the list mentioned above, see [https://discord.com/channels/960643023006490684/1239205785913790465/1456317703156531211 Discord]. Approximate score: 4e12. Later, he found another halting machine in the same list, see [https://discord.com/channels/960643023006490684/1239205785913790465/1456381492019069192 Discord] - approximate score: 1.5e18.
** @mxdys shared a list of machines that seem to be provable. See [https://discord.com/channels/960643023006490684/1460495597386731643/1460495597386731643 Discord].
** Dyuan [https://discord.com/channels/960643023006490684/1460495597386731643/1463044134263853066 shared an informal proof] for two machines nonhalting, using [[Longitudinal Analysis]].
** @Peacemaker II [https://discord.com/channels/960643023006490684/1239205785913790465/1466337165305974806 found] two machines to halt with a bespoke [[accelerated simulator]].
** @mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 released] a [[holdouts list]] of 1314 machines up to equivalence, which is a 1% reduction (12 machines) from last month. 4 of these were not previously simulated up to 1e13 steps, thus that count has been lowered to 206.
** @mxdys [https://discord.com/channels/960643023006490684/1460495597386731643/1466440885700006123 announced] the start of translating three proofs for halting machines into [[wikipedia:Rocq|Rocq]] (two machines found by @Peacemaker II this month, one proof for [[1RB0RC 0LC0LB 0LD1LC 0LE1LA 0LF--- 1RF1RA|a machine]] found to halt by Racheline in July 2024, which has previously been missed when accounting for the informal holdout count, therefore the informal holdout count is now one less).
** All machines of the "Unknown" class have been simulated up to 1e13 steps, with aparker simulating the last machine to 1.15e13: [https://discord.com/channels/960643023006490684/1458063260959113279/1468676578455064742 Discord].
*[[BB(7)]]:
**Andrew Ducharme further continued reducing the number of holdouts, from 20,387,509 to '''20,197,978''' TMs, a 0.93% reduction. This puts the total compute time on the current BB(7) pipeline just over '''50,000''' hours.[https://discord.com/channels/960643023006490684/1369339127652159509/1459719965396566078]
**The holdout count was further reduced to '''19,879,953''' (a 1.57% reduction), which breaks the 20 million barrier.[https://discord.com/channels/960643023006490684/1369339127652159509/1460718041577951365]
**Another reduction brought the holdout count down to '''19,781,295''', a 0.5% reduction, then '''19,303,801''', a 2.41% reduction.
*[[BB(2,5)]]:
**Andrew Ducharme '''[https://discord.com/channels/960643023006490684/1259770421046411285/1466208979511414885 found] a machine to be nonhalting via @mxdys's FAR decider''', which was [https://discord.com/channels/960643023006490684/1259770421046411285/1466331107279769736 confirmed in Rocq].
*[[BB(2,6)]]:
**Andrew Ducharme reduced the number of holdouts from 870,085 to '''867,008''', a 0.35% reduction, with more application of Enumerate.py.
**Using the newly released mxdys FAR decider, the holdout count was brought down to '''558,039''', a 35.6% reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1467385975704391897]

[[Category:This Month in Beaver Research|2026-01]]

TMBR: January 2026

2026-03-06T19:57:52Z

ADucharme: summary rephrasing

{{TMBRnav|December 2025|February 2026}}

This is the first edition of TMBR in 2026. This month, we saw a rare occurrence: a [[BB(2,5)]] TM was proven to be nonhalting, by Andrew Ducharme. The BB(6) and BB(7) holdout reductions were also notable. These were aided by a C++ implementation of FAR with a DFA generator released by mxdys on the 18th - see [https://discord.com/channels/960643023006490684/1239205785913790465/1462674863398326362 Discord].

== Champions ==

* nickdrozd discovered a new [[BLB|BLB(3,3)]] [[champion]] ({{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}) which blanks the tape after running for more than 1042,745 steps.
* creeperman7002 discovered <code>1TB1PA_1PC0PA_1TA0PD_---1TA</code>, a [[TT|TT(4,2)]] TM which runs for 48,186 steps and <code>1TA2PB3TB---_3TA1PB1TA1PA</code>, a [[TT|TT(2,4)]] TM which runs for more than 3.467*1015 steps.
* A new lower bound of <math>f_{\varepsilon_0 \omega^{\omega^{3}}}\left(4\right)</math> was computed for the [[Busy Beaver for lambda calculus#Champions|BBλ(91)]] champion.
* New champions were discovered for BBλ(61), BBλ(86), BBλ(90), BBλ(94), BBλ(96) and BBλ(100).

== Blog Posts ==

* 16 Jan 2026. Nick Drozd. [https://nickdrozd.github.io/2026/01/16/ai-lin-busy-beaver.html AI Reproduction of Lin's Busy Beaver Proof].
* 28 Jan 2026. John Tromp. [https://tromp.github.io/blog/2026/01/28/largest-number-revised The largest number representable in 64 bits - Revised].

== BB Adjacent ==

* [https://discord.com/channels/960643023006490684/1458010522967609425/1458010522967609425 Uniform-Action Busy Beavers] were introduced, and lower bounds have been given up to BBu(6) (with BBu(2)=6, BBu(3)=17 and BBu(4)=29)

== Methods ==
mxdys released C++ implementation of FAR with a DFA generator. See [https://discord.com/channels/960643023006490684/1239205785913790465/1462674863398326362 Discord].

== Theory ==
@ConePine [https://discord.com/channels/960643023006490684/960643023530762341/1458876275749032232 shared an idea] on how the value of BB(5) can be proved without enumerating Turing machines.

== Holdouts ==

{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!New Holdout Count
!Previous Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(2,5)]]
|74
|75
|1
|1.3%
|-
|[[BB(6)]]
|1314
|1326
|12
|0.9%
|-
|[[BB(7)]]
|19,303,801
|20,387,509
|1,083,708
|5.32%
|-
|[[BB(2,6)]]
|558,039
|870,085
|312,046
|35.86%
|}
* [[BB(6)]]: '''15''' solved machines. 1311 holdouts. 2 informally solved this month.
** Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire and aparker: see [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 spreadsheet]. Current count: '''205'''. (Total reduction: '''73''' (+1 later). 39 machines[https://discord.com/channels/960643023006490684/1456755170527543420/1456755170527543420 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1456757189787127971/1456757189787127971 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1456757980203585678/1456757980203585678 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1456758622762696815/1456758622762696815 <nowiki>[4]</nowiki>][https://discord.com/channels/960643023006490684/1456759118818971709/1456759118818971709 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1456761765688901724/1456761765688901724 <nowiki>[6]</nowiki>][https://discord.com/channels/960643023006490684/1456762231491657830/1456762231491657830 <nowiki>[7]</nowiki>][https://discord.com/channels/960643023006490684/1456310285462802587/1456760768472289321 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456932659740545217 <nowiki>[9]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456933097009320089 <nowiki>[10]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456933714629099593 <nowiki>[11]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1456934113943621653 <nowiki>[12]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457147385309565021 <nowiki>[13]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457409447084429594 <nowiki>[11 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457497643478683770 <nowiki>[4 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457644806984437760 <nowiki>[10 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1457992404597739672 <nowiki>[11 more]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1458413343088840734 <nowiki>[17 more]</nowiki>][https://discord.com/channels/960643023006490684/1458063260959113279/1468676578455064742 <nowiki>[aparker's machine]</nowiki>] simulated out, plus 6 solved machines).
** Alistaire found a halting machine in the list mentioned above, see [https://discord.com/channels/960643023006490684/1239205785913790465/1456317703156531211 Discord]. Approximate score: 4e12. Later, he found another halting machine in the same list, see [https://discord.com/channels/960643023006490684/1239205785913790465/1456381492019069192 Discord] - approximate score: 1.5e18.
** @mxdys shared a list of machines that seem to be provable. See [https://discord.com/channels/960643023006490684/1460495597386731643/1460495597386731643 Discord].
** Dyuan [https://discord.com/channels/960643023006490684/1460495597386731643/1463044134263853066 shared an informal proof] for two machines nonhalting, using [[Longitudinal Analysis]].
** @Peacemaker II [https://discord.com/channels/960643023006490684/1239205785913790465/1466337165305974806 found] two machines to halt with a bespoke [[accelerated simulator]].
** @mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 released] a [[holdouts list]] of 1314 machines up to equivalence, which is a 1% reduction (12 machines) from last month. 4 of these were not previously simulated up to 1e13 steps, thus that count has been lowered to 206.
** @mxdys [https://discord.com/channels/960643023006490684/1460495597386731643/1466440885700006123 announced] the start of translating three proofs for halting machines into [[wikipedia:Rocq|Rocq]] (two machines found by @Peacemaker II this month, one proof for [[1RB0RC 0LC0LB 0LD1LC 0LE1LA 0LF--- 1RF1RA|a machine]] found to halt by Racheline in July 2024, which has previously been missed when accounting for the informal holdout count, therefore the informal holdout count is now one less).
** All machines of the "Unknown" class have been simulated up to 1e13 steps, with aparker simulating the last machine to 1.15e13: [https://discord.com/channels/960643023006490684/1458063260959113279/1468676578455064742 Discord].
*[[BB(7)]]:
**Andrew Ducharme further continued reducing the number of holdouts, from 20,387,509 to '''20,197,978''' TMs, a 0.93% reduction. This puts the total compute time on the current BB(7) pipeline just over '''50,000''' hours.[https://discord.com/channels/960643023006490684/1369339127652159509/1459719965396566078]
**The holdout count was further reduced to '''19,879,953''' (a 1.57% reduction), which breaks the 20 million barrier.[https://discord.com/channels/960643023006490684/1369339127652159509/1460718041577951365]
**Another reduction brought the holdout count down to '''19,781,295''', a 0.5% reduction, then '''19,303,801''', a 2.41% reduction.
*[[BB(2,5)]]:
**Andrew Ducharme '''[https://discord.com/channels/960643023006490684/1259770421046411285/1466208979511414885 found] a machine to be nonhalting via @mxdys's FAR decider''', which was [https://discord.com/channels/960643023006490684/1259770421046411285/1466331107279769736 confirmed in Rocq].
*[[BB(2,6)]]:
**Andrew Ducharme reduced the number of holdouts from 870,085 to '''867,008''', a 0.35% reduction, with more application of Enumerate.py.
**Using the newly released mxdys FAR decider, the holdout count was brought down to '''558,039''', a 35.6% reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1467385975704391897]

[[Category:This Month in Beaver Research|2026-01]]

TMBR: February 2026

2026-03-01T01:37:42Z

ADucharme: typo + change "newly released mxdys decider" to "mxdys C++ FAR decider"

User:ADucharme

2026-02-27T20:34:21Z

ADucharme: /* BB(6) */ add solved BB6 TMs

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts since that enumeration. I've also tried my hand at the analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last ~1500 BB(6) holdouts, I solved 66 and counting. Partial credit for some of these machines goes to Peacemaker II, who identifies permutations of machines I solved in the holdout list. Because of the often shared behavior between permutations, I can apply the decider which solved to the original TM I found to the permutations, and often solve all the permutations too.

Solved halting TMs (49) with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs (19) with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9
1RB0RB_1LC1RA_0LA1RD_1LA1LE_1LF1LD_---0LC FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 3 n 12
1RB0LD_1RC0RE_0LA0RC_1LA1LD_0RF1RA_---1RC FAR CPS_LRU maxT 10000000 LRUH 6 H 3 tH 0 n 9
1RB1LB_1LC1RE_0RD0LB_0LB1RA_1LA0RF_---0RC FAR CPS_LRU maxT 10000000 LRUH 7 H 3 tH 1 n 4
1RB0LD_0RC1RF_1RD0RA_1LE1RB_1LC0LE_1RC--- FAR CPS_LRU maxT 10000000 LRUH 7 H 4 tH 1 n 24
1RB0LA_0RC---_1RD1RE_1LA1LD_1RD0RF_0RC1RC FAR RWL_mod maxT 10000000 H 8 mod 3 n 6
1RB0LA_1RC1RA_0LD1LA_1LF1RE_0RD0RE_0LC--- FAR RWL_mod maxT 10000000 H 4 mod 1 n 8
1RB1RF_1LC1LB_---0LD_1RE0LD_0RA1RA_0LE0RE FAR RWL_mod maxT 10000000 H 8 mod 3 n 6

These non-halters are permutations of a TM I solved. The permutations were identified by Peacemaker II, but mxdys was the one who actually applied the decider. These also required flipping every instruction's direction of travel to work.
1LB0RD_1RC1LC_1LE1RA_---0RE_0RF0LC_0LC1RB FAR CPS_LRU maxT 1000000 LRUH 7 H 3 tH 1 n 4
1LB1RE_0RC0LA_0LA1RD_1RA1LA_1LD0RF_---0RB FAR CPS_LRU maxT 1000000 LRUH 7 H 3 tH 1 n 4

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Games ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain. These remain my best-ever entries in these particular Busy Beaver games.

==== BB(7) ====
{| class="wikitable sortable"
|Place
|TM
|Score
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|5
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG}}
|10 ↑↑ 403.84
|-
|9
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}}
|10 ↑↑ 243.88
|}

{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}} was a bit of co-discovery: Iijil first enumerated the TM and I first showed it was halting.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}

User:ADucharme

2026-02-22T19:50:13Z

ADucharme: /* BB(6) */ credit Peacemaker, add LRUH 7 permutations

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts since that enumeration. I've also tried my hand at the analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last ~1500 BB(6) holdouts, I solved 63 and counting. Partial credit for some of these machines goes to Peacemaker II, who identifies permutations of machines I solved in the holdout list. Because of the often shared behavior between permutations, I can apply the decider which solved to the original TM I found to the permutations, and often solve all the permutations too.

Solved halting TMs (49) with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs (16) with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9
1RB0RB_1LC1RA_0LA1RD_1LA1LE_1LF1LD_---0LC FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 3 n 12
1RB0LD_1RC0RE_0LA0RC_1LA1LD_0RF1RA_---1RC FAR CPS_LRU maxT 10000000 LRUH 6 H 3 tH 0 n 9
1RB1LB_1LC1RE_0RD0LB_0LB1RA_1LA0RF_---0RC FAR CPS_LRU maxT 10000000 LRUH 7 H 3 tH 1 n 4
1RB0LD_0RC1RF_1RD0RA_1LE1RB_1LC0LE_1RC--- FAR CPS_LRU maxT 10000000 LRUH 7 H 4 tH 1 n 24
These non-halters are permutations of a TM I solved. The permutations were identified by Peacemaker II, but mxdys was the one who actually applied the decider. These also required flipping every instruction's direction of travel to work.
1LB0RD_1RC1LC_1LE1RA_---0RE_0RF0LC_0LC1RB FAR CPS_LRU maxT 1000000 LRUH 7 H 3 tH 1 n 4
1LB1RE_0RC0LA_0LA1RD_1RA1LA_1LD0RF_---0RB FAR CPS_LRU maxT 1000000 LRUH 7 H 3 tH 1 n 4

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Games ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain. These remain my best-ever entries in these particular Busy Beaver games.

==== BB(7) ====
{| class="wikitable sortable"
|Place
|TM
|Score
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|5
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG}}
|10 ↑↑ 403.84
|-
|9
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}}
|10 ↑↑ 243.88
|}

{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}} was a bit of co-discovery: Iijil first enumerated the TM and I first showed it was halting.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}

User:ADucharme

2026-02-20T07:21:17Z

ADucharme: /* Holdout Reduction */ add nonhalting bb6 tms

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts since that enumeration. I've also tried my hand at the analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last ~1500 BB(6) holdouts, I solved 61 and counting.

Solved halting TMs (49) with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs (12) with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9
1RB0RB_1LC1RA_0LA1RD_1LA1LE_1LF1LD_---0LC FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 3 n 12
1RB0LD_1RC0RE_0LA0RC_1LA1LD_0RF1RA_---1RC FAR CPS_LRU maxT 10000000 LRUH 6 H 3 tH 0 n 9
1RB1LB_1LC1RE_0RD0LB_0LB1RA_1LA0RF_---0RC FAR CPS_LRU maxT 10000000 LRUH 7 H 3 tH 1 n 4
1RB0LD_0RC1RF_1RD0RA_1LE1RB_1LC0LE_1RC--- FAR CPS_LRU maxT 10000000 LRUH 7 H 4 tH 1 n 24

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Games ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain. These remain my best-ever entries in these particular Busy Beaver games.

==== BB(7) ====
{| class="wikitable sortable"
|Place
|TM
|Score
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|5
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG}}
|10 ↑↑ 403.84
|-
|9
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}}
|10 ↑↑ 243.88
|}

{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}} was a bit of co-discovery: Iijil first enumerated the TM and I first showed it was halting.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}

User:ADucharme

2026-02-14T02:42:01Z

ADucharme: /* Holdout Reduction */ add BB(6) solved non-halters

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts since that enumeration. I've also tried my hand at the analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last 2592 BB(6) holdouts, I solved 59 (2.28%) and counting.

Solved halting TMs (49) with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs (10) with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9
1RB0RB_1LC1RA_0LA1RD_1LA1LE_1LF1LD_---0LC FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 3 n 12
1RB0LD_1RC0RE_0LA0RC_1LA1LD_0RF1RA_---1RC FAR CPS_LRU maxT 10000000 LRUH 6 H 3 tH 0 n 9

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Games ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain. These remain my best-ever entries in these particular Busy Beaver games.

==== BB(7) ====
{| class="wikitable sortable"
|Place
|TM
|Score
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|5
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG}}
|10 ↑↑ 403.84
|-
|9
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}}
|10 ↑↑ 243.88
|}

{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}} was a bit of co-discovery: Iijil first enumerated the TM and I first showed it was halting.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}

User:ADucharme

2026-02-13T04:11:34Z

ADucharme: /* Busy Beaver Games */ templating

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts since that enumeration. I've also tried my hand at the analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last 2592 BB(6) holdouts, I solved 57 (2.19%) and counting.

Solved halting TMs with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6 run
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6 run
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6 run
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20 run
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9 run

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Games ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain. These remain my best-ever entries in these particular Busy Beaver games.

==== BB(7) ====
{| class="wikitable sortable"
|Place
|TM
|Score
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|T-2
|{{TM|1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE}}
|10 ↑↑ 519.20
|-
|5
|{{TM|1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG}}
|10 ↑↑ 403.84
|-
|9
|{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}}
|10 ↑↑ 243.88
|}

{{TM|1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG}} was a bit of co-discovery: Iijil first enumerated the TM and I first showed it was halting.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}

User:ADucharme

2026-02-13T04:09:50Z

ADucharme: /* Busy Beaver Game */ add bb7 results

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts since that enumeration. I've also tried my hand at the analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last 2592 BB(6) holdouts, I solved 57 (2.19%) and counting.

Solved halting TMs with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6 run
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6 run
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6 run
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20 run
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9 run

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Games ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain. These remain my best-ever entries in these particular Busy Beaver games.

==== BB(7) ====
{| class="wikitable sortable"
|Place
|TM
|Score
|-
|T-2
|<code>[[1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LF_0RD1LF_1LB0LE]]</code> (bbch)
|10 ↑↑ 519.20
|-
|T-2
|<code>1RB1RZ_0RC0RE_1LD1LA_1LC0LG_0RF1LE_0RD1LF_1LB0LE</code> (bbch)
|10 ↑↑ 519.20
|-
|5
|<code>1RB1LB_1LC1RF_1LA0LD_1RE0LG_0RC1RZ_0RB0RD_0RF1LG</code> (bbch)
|10 ↑↑ 403.84
|-
|9
|<code>1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG</code> (bbch)
|10 ↑↑ 243.88
|}
{| class="wikitable sortable"
|<code>1RB1RZ_1RC0LE_0RD1RB_1LE1RA_1LF0LG_0LG0RG_1LB1RG</code> (bbch)
|}
was a bit of co-discovery: Iijil first enumerated the TM and I showed it was halting.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}

BB(6)

2026-02-13T04:03:22Z

ADucharme: fix top score formatting

The 6-state, 2-symbol Busy Beaver problem, '''BB(6)''', refers to the unsolved 6th 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_1RC1RZ_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>

== History ==
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
@mxdys's informal [[Holdouts lists|holdouts list]] has 1314 machines up to equivalence as of January 2026. Partial Rocq proof is [https://github.com/ccz181078/busycoq/tree/BB6 available on Github].

Always up-to-date annotated spreadsheet, with links to Discord discussions: [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Spreadsheet].

All holdouts have been simulated up to at least 1e12 steps. 210 holdouts have not been simulated out to 1e13 steps. (The list can be found in the spreadsheet)
== 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
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}

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}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

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|1RB0LD_0RC0RE_1LC0LA_---1LE_0LF0RE_0RA0LE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 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 [[wikipedia:Knuth's_up-arrow_notation|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
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|-
|{{TM|1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA|halt}}
|<math>> 10^{11\,952\,340}</math> (lower bound)
|Racheline
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.
== 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]]).

We are also applying existing automatic deciders on current holdout lists with more extreme choices of parameters (more computational resources). [[User:XnoobSpeakable|XnoobSpeakable]] was able to solve 11 of the final 2728 holdouts using higher order parameters with the Ligockis' Enumerate.py. An example command line entry is:
<syntaxhighlight lang="bash">
python3 Code/Enumerate.py --infile "bb6in/bb6tm{i}.txt" --outfile "bb6out/t{i}.pb" -r --no-steps --exp-linear-rules --max-loops=50_000_000 --block-mult=3 --max-block-size=100 --time=500 --force --save-freq=1
</syntaxhighlight>
XnoobSpeakable ran Enumerate.py on all TMs in the 2728 holdout list with the above max-loops and max-block-size parameters using <code>--block-mult=1</code> ,<code>--block-mult=2</code> , and <code>--block-mult=3</code>. For context, during the Stage 2 BB(7) enumeration, where speed was more important due to the tens of millions of known holdouts, parameters of <code>--max-loops=100_000 --block-mult=2 --time=30 --save-freq=100</code> were used.

@Iijil's [[MITMWFAR|MITMWFAR decider]] is likely too weak to be of any assistance: running the decider on 2650 BB(6) holdouts, using parameters not strong enough to solve BB(5) TMs, took prohibitively long to compute. Instead, [https://discord.com/channels/960643023006490684/1028746861395316776/1442964185599447152 a new FAR method] by mxdys was able to decide 113 of the 1534 holdouts ([https://github.com/ccz181078/TM/tree/FAR code] on GitHub).

== References ==
<references />

[[Category:BB Domains]][[Category:BB(6)]]

BB(6)

2026-02-13T04:01:40Z

ADucharme: reorder sections to match other domains like 2x5, 3x3, 2x6, 7x2

User:ADucharme

2026-02-13T03:55:02Z

ADucharme: /* Busy Beaver Game */ getting TM modulized

Hi, I'm Andrew!

My main contribution to bbchallenge is applying the Ligocki and mxdys deciders to many of the next unsolved domains. I helped organize the initial BB(7) enumeration and solved over 50% of all holdouts since that enumeration. I've also tried my hand at the analysis of some TMs, most notably BMO #1 and the Bonus Cryptid, but have not ever solved a TM by hand. Below are the TMs I've solved for the most actively studied BB domains.

== Holdout Reduction ==

==== BB(6) ====
Of the last 2592 BB(6) holdouts, I solved 57 (2.19%) and counting.

Solved halting TMs with sigma score
1RB---_1LC0LA_1LD0RD_0RE0LB_1RC1RF_0RD1RF ~10^79.95448
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD--- ~10^70.05261
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD--- ~10^70.00750
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA---
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.99803
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC---
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC---
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC--- ~10^69.94652
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC0RF_1LE--- ~10^52.44977
1RB1LA_0LB1LC_1RD0LD_0LA0RE_1RC1RF_0LD---
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1LF_---1RA ~10^52.25998
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0RC1RE_0RC---
1RB0RD_1RC1RA_1LD1LA_0RE0LC_0LF1RF_0RB--- ~10^38.85754
1RB0RD_1RC1RA_1LD1LA_0RE0LC_1RC1RF_0RB1RZ
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC1RF_0LD0LA 3_804_764_807_033_118_405_271_455_910_658_686_671_560_877_296_302
1RB---_1LC1LF_1RD0LD_0LB0RE_1RC0RE_0RF0LA
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA0RF 2_802_749_143_558_201_797_723_325_357_510_324_775_865_733_035_298
1RB---_1RC0LC_0LD0RF_1LB1LE_0LC1LE_1RB0RA 224_322_871_042_507_036_371_085_207_200_624_692_576_495_497_310
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD---
1RB---_1RC0LC_0LD0RF_1RE1LD_0LE1LB_1RB0RA
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0RC1RZ 87_112_055_695_139_218_500_268_260_804_164_378
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LF_0LE1RF_0RC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0RC1RF_0LA1RZ
1RB1RE_1LC0RC_0RA0LD_1LB0LD_0LE1RF_0RC1RZ
1RB1RF_1RC0LC_0LD0RA_1LB1LE_0LC1LF_0LC1RZ
1RB1RE_1LC0RC_0RA0LD_1LB1LE_0RC1RF_0LA1RZ
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0RE0LF_1RD1RZ 87_112_055_695_139_218_500_268_260_804_164_377
1RB0RA_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB1RE_1RC0LC_0LD0RA_1LB1LE_0LC0LF_1RD1RZ
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1LE--- 708_804_434_842_666_889_215_481_456_393_612
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1RF_1RA1RD_---1LC 5_652_984_156_355_601_606_126_039_264
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1RA---
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA0RF_1LE---
1RB0LB_0LC0RE_1LA1LD_0LB0LF_1RA0RE_1RC---
1RB0LB_0LC0RF_1LA1LD_0RD1LE_0LB---_1RA1RE
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0RC---
1RB0LB_0LC0RE_1RD1LC_0LD1LA_1RA0RF_1RA--- 24_585_555_916_266_386_719_525
1RB0LB_0LC0RE_1LA1LD_0LB1LD_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0RC---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RF_0LB---
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA0RE_0LB--- 12_878_567_902_665_915
1RB0LB_0LC0RE_1LA1LD_0LB1LF_1RA1RD_0LB---
1RB1LA_1LC0RC_1LD1RC_1LD1LE_0LF0LA_1RZ0RA 19,694
1RB1LA_1LC0RC_1LD1RC_0LC1LE_0LF0LA_---0RA
Solved non-halting TMs with decider
1RB1RF_1LC0RD_1RE0RD_0RC0LE_1LB0RA_0RE--- Inf Proof_System
1RB0LF_0RC0RF_1RD---_1LE0LB_1LA0LD_1RA0RE Inf Proof_System
1RB0LE_1LC0LE_1RA0LD_1LA1LF_0LB0RC_0LC--- Inf Proof_System
1RB0RF_1RC---_1RD1LF_1RE0RD_0LC1RA_1LC0LF FAR CPS_LRU maxT 10000000 LRUH 4 H 2 tH 0 n 6 run
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_1RA--- FAR CPS_LRU maxT 10000000 LRUH 4 H 4 tH 0 n 6 run
1RB1LD_1RC0RB_0LA1RE_1LA0LD_1RF0RD_0RC--- FAR CPS_LRU maxT 10000000 LRUH 4 H 3 tH 0 n 6 run
1RB1LA_0RC0LF_0RD---_1RE1RD_1LB1RA_0LD0LA FAR CPS_LRU maxT 10000000 LRUH 1 H 1 tH 0 n 20 run
1RB0RB_1LC0LE_0RF1LD_1RA0LB_1RA0RD_---0RC FAR CPS_LRU maxT 10000000 LRUH 4 H 1 tH 3 n 9 run

==== BB(2,5) ====
Of the last 75 2x5 holdouts, I have solved 2 (2.68%).

Solved non-halting TM with decider
1RB2LA0RB1LB0LB_1LA3RA1RA4RA--- FAR CPS_LRU maxT 10000000 LRUH 6 H 1 tH 0 n 2
1RB2RB---0LB3LA_2LA2LB3RB4RB1LB FAR CPS_LRU maxT 10000000 LRUH 8 H 5 tH 0 n 2

== Busy Beaver Game ==
Through my filtering, I've compiled a few of the highest-scoring halters for several domains. I've never taken first place, but I've come close. If only uni would make his code public...

This section lists any TMs in the current top 10 for a given domain in addition to my personal all-time best.

==== BB(2,6) ====
{| class="wikitable"
|Place
|TM
|Score
|-
|6
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|-
|7
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|-
|8
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|}