BusyBeaverWiki - User contributions [en]

User:RobinCodes/Work on BB Domains

2026-05-10T06:02:19Z

RobinCodes: Add BB(3,4) as active domain and add progress

== Progress ==
As of May 2026,

'''Domains with active filtering going on (decider-wise): [[BB(2,6)]], [[BB(7)]], [[BB(4,3)]] - Andrew Ducharme. [[BB(4,3)|BB(3,4)]] - [[User:XnoobSpeakable|XnoobSpeakable]]'''

{| class="wikitable"
|+ Work on BB Domains
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
!8-state
!9-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
2 Jul 2024
| style="background: #A8613E;" | 1104
[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Holdouts list]

(2 informal)

16+6 Cryptids

[[BB(6)]]
| style="background: #A06B10;" | 17,823,260
No Handwork
Andrew Ducharme, Terry Ligocki
Phase 2, Stage 6

[[BB(7)]]
| style="background: #AA8519;" |Expected no. of holdouts: 147 billion (<math>1.47 \cdot 10^{11}</math>)
[[BB(8)]]
| style="background: #ECC652;" |
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: #A8613E;" | 4
[[BB(3,3)#Holdouts|Holdouts list]]

1 Cryptid, Exhausted

[[BB(3,3)]]
| style="background: #A06B10;" | 5,127,263
No Handwork

Terry Ligocki, Andrew Ducharme Phase 2, Stage 3

[[BB(4,3)]]
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #A8973E;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #A06B10;" | 12,049,358
No Handwork
[[User:XnoobSpeakable|XnoobSpeakable,]] [[User:WarpedWartWars|Lúkos]], Andrew Ducharme

Phase 3

[[BB(3,4)]]
| style="background: #D8B446;" |Expected no. of holdouts: 1.53 quadrillion (<math>1.53 \cdot 10^{15}</math>)
[[BB(4,4)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
! 5-symbol
| style="background: #A8613E;" | 60
[[BB(2,5)#Holdouts|Holdouts list]]
(6 informal)

2+4 Cryptids

[[BB(2,5)]]
| style="background: #D8B446;" |Lower bound: (due Racheline 2024)<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
Expected no. of holdouts: 11.3 trillion (<math>1.13 \cdot 10^{13}</math>)
[[BB(3,5)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
! 6-symbol
| style="background: #A06B10;" | 439,120
No Handwork
Andrew Ducharme,
Terry Ligocki

Phase 2 Stage 4

[[BB(2,6)]]
| style="background: #ECC652;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
!7-symbol
| style="background: #AA8519;" |Enumeration started!
24% Done

Expected no. of holdouts: 3 billion (<math>3 \cdot 10^9</math>)

Current: 750 million

[[BB(2,7)]]
| style="background: #ECC652;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
!8-symbol
| style="background: #ECC652;" |Expected no. of holdouts: 5.51 trillion (<math display="inline">5.51 \cdot 10^{12}</math>)
[[BB(2,8)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|}
Futher domains with some information:

* [[BB(11)]]: Lower bound: (Racheline 2024)
<math display="block">f_\omega^2(2 \uparrow\uparrow 12)</math>

* [[BB(12)]]: Lower bound: (Racheline 2024)
<math display="block">f_\omega^4(2 \uparrow\uparrow\uparrow 4-3)</math>
== Enumeration ==
The enumeration process for [[BB(2,7)]] has started, and 10% has so far been completed. [[BB(8)]] is (tested from a random sample) even harder to enumerate. These processes will take a very long time and much collaborative work to complete. As for BB(3x5), BB(4x4), BB(5x3), enumeration may still be possible, but also incredibly difficult. Any domains after these are probably complete out-of-reach of current technology, just to enumerate. Using crowdsourced enumeration, one more layer may be possible.

== Informal Proofs ==
'''Unconfirmed, informal proofs of machines for each domain by equivalence class: 8 total'''
{| class="wikitable"
!Domain
!2-state
!3-state
!4-state
!5-state
!6-state
|-
!2-symbol
|0
|0
|0
|0
|'''2'''
|-
!3-symbol
|0
|0
| ----
| ---
| ---
|-
!4-symbol
|0
| ---
| ---
| ---
| ---
|-
!5-symbol
|'''6'''
| ---
| ---
| ---
| ---
|}
=== [[BB(6)]]: 1 March 2026 ===

===== 2 informal =====

*mxdys, Pomme, Autumn Pan, vyx[https://discord.com/channels/960643023006490684/1239205785913790465/1441124403801755730 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1443295684878143579 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1444047803935490190 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1444051070186819836 <nowiki>[4]</nowiki>]: <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code>
*<code>1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA</code> (halt, Racheline, informal language)

=== [[BB(2,5)]]: 1 April 2026 ===

===== 5+1 informal =====

* Dyuan:
*# Dekaheptoid, unverified: [[1RB3RB1LB---2RB_2LA1RA4LB2LA2RA]] ([https://discord.com/channels/960643023006490684/1259770421046411285/1267650177389432913 source]), equivalently (1.2) <code>1RB3RA2LB1LB1RB_2LA2RA4LA1LA---</code>
* Legion:
*# <code>1RB2RA3LA4LA2RB_2LA---1LA1RA3RA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 1])
*# <code>1RB3LA4LA1LA2RA_2LA4RB---0RA0LA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 2])
*# <code>1RB3LA4LA2RB1LA_2LA4RB---3RA3LA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 3])
* [https://discord.com/channels/960643023006490684/1259770421046411285/1329663999700111471 Racheline]: <code>1RB2LB---4LB0RB_1LA3RB4RB4RA1LB</code>
=== [[BB(3,3)]]: 28 March 2026 ===
''None''

User:RobinCodes/Work on BB Domains

2026-05-10T05:58:28Z

RobinCodes: add (4,3) to active domains, add (2,6) progress

== Progress ==
As of May 2026,

'''Domains with active filtering going on (decider-wise): [[BB(2,6)]], [[BB(7)]] - Andrew Ducharme. [[BB(4,3)]] - Andrew Ducharme & [[User:XnoobSpeakable|XnoobSpeakable]]'''

{| class="wikitable"
|+ Work on BB Domains
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
!8-state
!9-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
2 Jul 2024
| style="background: #A8613E;" | 1104
[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Holdouts list]

(2 informal)

16+6 Cryptids

[[BB(6)]]
| style="background: #A06B10;" | 17,823,260
No Handwork
Andrew Ducharme, Terry Ligocki
Phase 2, Stage 6

[[BB(7)]]
| style="background: #AA8519;" |Expected no. of holdouts: 147 billion (<math>1.47 \cdot 10^{11}</math>)
[[BB(8)]]
| style="background: #ECC652;" |
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: #A8613E;" | 4
[[BB(3,3)#Holdouts|Holdouts list]]

1 Cryptid, Exhausted

[[BB(3,3)]]
| style="background: #A06B10;" | 5,641,006
No Handwork

Terry Ligocki, Andrew Ducharme Phase 2, Stage 3

[[BB(4,3)]]
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #A8973E;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #A06B10;" | 12,049,358
No Handwork
[[User:XnoobSpeakable|XnoobSpeakable,]] [[User:WarpedWartWars|Lúkos]], Andrew Ducharme

Phase 3

[[BB(3,4)]]
| style="background: #D8B446;" |Expected no. of holdouts: 1.53 quadrillion (<math>1.53 \cdot 10^{15}</math>)
[[BB(4,4)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
! 5-symbol
| style="background: #A8613E;" | 60
[[BB(2,5)#Holdouts|Holdouts list]]
(6 informal)

2+4 Cryptids

[[BB(2,5)]]
| style="background: #D8B446;" |Lower bound: (due Racheline 2024)<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
Expected no. of holdouts: 11.3 trillion (<math>1.13 \cdot 10^{13}</math>)
[[BB(3,5)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
! 6-symbol
| style="background: #A06B10;" | 439,120
No Handwork
Andrew Ducharme,
Terry Ligocki

Phase 2 Stage 4

[[BB(2,6)]]
| style="background: #ECC652;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
!7-symbol
| style="background: #AA8519;" |Enumeration started!
24% Done

Expected no. of holdouts: 3 billion (<math>3 \cdot 10^9</math>)

Current: 750 million

[[BB(2,7)]]
| style="background: #ECC652;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
!8-symbol
| style="background: #ECC652;" |Expected no. of holdouts: 5.51 trillion (<math display="inline">5.51 \cdot 10^{12}</math>)
[[BB(2,8)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|}
Futher domains with some information:

* [[BB(11)]]: Lower bound: (Racheline 2024)
<math display="block">f_\omega^2(2 \uparrow\uparrow 12)</math>

* [[BB(12)]]: Lower bound: (Racheline 2024)
<math display="block">f_\omega^4(2 \uparrow\uparrow\uparrow 4-3)</math>
== Enumeration ==
The enumeration process for [[BB(2,7)]] has started, and 10% has so far been completed. [[BB(8)]] is (tested from a random sample) even harder to enumerate. These processes will take a very long time and much collaborative work to complete. As for BB(3x5), BB(4x4), BB(5x3), enumeration may still be possible, but also incredibly difficult. Any domains after these are probably complete out-of-reach of current technology, just to enumerate. Using crowdsourced enumeration, one more layer may be possible.

== Informal Proofs ==
'''Unconfirmed, informal proofs of machines for each domain by equivalence class: 8 total'''
{| class="wikitable"
!Domain
!2-state
!3-state
!4-state
!5-state
!6-state
|-
!2-symbol
|0
|0
|0
|0
|'''2'''
|-
!3-symbol
|0
|0
| ----
| ---
| ---
|-
!4-symbol
|0
| ---
| ---
| ---
| ---
|-
!5-symbol
|'''6'''
| ---
| ---
| ---
| ---
|}
=== [[BB(6)]]: 1 March 2026 ===

===== 2 informal =====

*mxdys, Pomme, Autumn Pan, vyx[https://discord.com/channels/960643023006490684/1239205785913790465/1441124403801755730 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1443295684878143579 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1444047803935490190 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1444051070186819836 <nowiki>[4]</nowiki>]: <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code>
*<code>1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA</code> (halt, Racheline, informal language)

=== [[BB(2,5)]]: 1 April 2026 ===

===== 5+1 informal =====

* Dyuan:
*# Dekaheptoid, unverified: [[1RB3RB1LB---2RB_2LA1RA4LB2LA2RA]] ([https://discord.com/channels/960643023006490684/1259770421046411285/1267650177389432913 source]), equivalently (1.2) <code>1RB3RA2LB1LB1RB_2LA2RA4LA1LA---</code>
* Legion:
*# <code>1RB2RA3LA4LA2RB_2LA---1LA1RA3RA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 1])
*# <code>1RB3LA4LA1LA2RA_2LA4RB---0RA0LA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 2])
*# <code>1RB3LA4LA2RB1LA_2LA4RB---3RA3LA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 3])
* [https://discord.com/channels/960643023006490684/1259770421046411285/1329663999700111471 Racheline]: <code>1RB2LB---4LB0RB_1LA3RB4RB4RA1LB</code>
=== [[BB(3,3)]]: 28 March 2026 ===
''None''

User:RobinCodes/Work on BB Domains

2026-05-06T06:10:37Z

RobinCodes: /* Progress */ Add BB(2,6) progress

== Progress ==
As of May 2026,

'''Domains with active filtering going on (decider-wise): [[BB(2,6)]], [[BB(7)]] - Andrew Ducharme'''

{| class="wikitable"
|+ Work on BB Domains
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
!8-state
!9-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
2 Jul 2024
| style="background: #A8613E;" | 1104
[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Holdouts list]

(2 informal)

16+6 Cryptids

[[BB(6)]]
| style="background: #A06B10;" | 17,823,260
No Handwork
Andrew Ducharme, Terry Ligocki
Phase 2, Stage 6

[[BB(7)]]
| style="background: #AA8519;" |Expected no. of holdouts: 147 billion (<math>1.47 \cdot 10^{11}</math>)
[[BB(8)]]
| style="background: #ECC652;" |
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: #A8613E;" | 4
[[BB(3,3)#Holdouts|Holdouts list]]

1 Cryptid, Exhausted

[[BB(3,3)]]
| style="background: #A06B10;" | 5,641,006
No Handwork

Terry Ligocki, Andrew Ducharme Phase 2, Stage 3

[[BB(4,3)]]
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #A8973E;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #A06B10;" | 12,049,358
No Handwork
[[User:XnoobSpeakable|XnoobSpeakable,]] [[User:WarpedWartWars|Lúkos]], Andrew Ducharme

Phase 3

[[BB(3,4)]]
| style="background: #D8B446;" |Expected no. of holdouts: 1.53 quadrillion (<math>1.53 \cdot 10^{15}</math>)
[[BB(4,4)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
! 5-symbol
| style="background: #A8613E;" | 60
[[BB(2,5)#Holdouts|Holdouts list]]
(6 informal)

2+4 Cryptids

[[BB(2,5)]]
| style="background: #D8B446;" |Lower bound: (due Racheline 2024)<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
Expected no. of holdouts: 11.3 trillion (<math>1.13 \cdot 10^{13}</math>)
[[BB(3,5)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
! 6-symbol
| style="background: #A06B10;" | 527,232
No Handwork
Andrew Ducharme,
Terry Ligocki

Phase 2 Stage 4

[[BB(2,6)]]
| style="background: #ECC652;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
!7-symbol
| style="background: #AA8519;" |Enumeration started!
24% Done

Expected no. of holdouts: 3 billion (<math>3 \cdot 10^9</math>)

Current: 750 million

[[BB(2,7)]]
| style="background: #ECC652;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
!8-symbol
| style="background: #ECC652;" |Expected no. of holdouts: 5.51 trillion (<math display="inline">5.51 \cdot 10^{12}</math>)
[[BB(2,8)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|}
Futher domains with some information:

* [[BB(11)]]: Lower bound: (Racheline 2024)
<math display="block">f_\omega^2(2 \uparrow\uparrow 12)</math>

* [[BB(12)]]: Lower bound: (Racheline 2024)
<math display="block">f_\omega^4(2 \uparrow\uparrow\uparrow 4-3)</math>
== Enumeration ==
The enumeration process for [[BB(2,7)]] has started, and 10% has so far been completed. [[BB(8)]] is (tested from a random sample) even harder to enumerate. These processes will take a very long time and much collaborative work to complete. As for BB(3x5), BB(4x4), BB(5x3), enumeration may still be possible, but also incredibly difficult. Any domains after these are probably complete out-of-reach of current technology, just to enumerate. Using crowdsourced enumeration, one more layer may be possible.

== Informal Proofs ==
'''Unconfirmed, informal proofs of machines for each domain by equivalence class: 8 total'''
{| class="wikitable"
!Domain
!2-state
!3-state
!4-state
!5-state
!6-state
|-
!2-symbol
|0
|0
|0
|0
|'''2'''
|-
!3-symbol
|0
|0
| ----
| ---
| ---
|-
!4-symbol
|0
| ---
| ---
| ---
| ---
|-
!5-symbol
|'''6'''
| ---
| ---
| ---
| ---
|}
=== [[BB(6)]]: 1 March 2026 ===

===== 2 informal =====

*mxdys, Pomme, Autumn Pan, vyx[https://discord.com/channels/960643023006490684/1239205785913790465/1441124403801755730 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1443295684878143579 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1444047803935490190 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1444051070186819836 <nowiki>[4]</nowiki>]: <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code>
*<code>1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA</code> (halt, Racheline, informal language)

=== [[BB(2,5)]]: 1 April 2026 ===

===== 5+1 informal =====

* Dyuan:
*# Dekaheptoid, unverified: [[1RB3RB1LB---2RB_2LA1RA4LB2LA2RA]] ([https://discord.com/channels/960643023006490684/1259770421046411285/1267650177389432913 source]), equivalently (1.2) <code>1RB3RA2LB1LB1RB_2LA2RA4LA1LA---</code>
* Legion:
*# <code>1RB2RA3LA4LA2RB_2LA---1LA1RA3RA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 1])
*# <code>1RB3LA4LA1LA2RA_2LA4RB---0RA0LA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 2])
*# <code>1RB3LA4LA2RB1LA_2LA4RB---3RA3LA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 3])
* [https://discord.com/channels/960643023006490684/1259770421046411285/1329663999700111471 Racheline]: <code>1RB2LB---4LB0RB_1LA3RB4RB4RA1LB</code>
=== [[BB(3,3)]]: 28 March 2026 ===
''None''

BB(2,6)

2026-05-06T06:10:08Z

RobinCodes: /* References */ Add progress with some TODOs

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="30" |[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
|-
|536,112
|533,764
|0.43%
|8778.5
| ---
| ---
|Enumerate.py -r --no-steps --exp-linear-rules --max-loops=100_000_000 --block-mult=7 --tape-limit=50_000 --max-steps-per-macro=1_000_000 --time=60 --lin-steps=0 --no-ctl
|-
|533,764
|527,232
|1.22%
|TODO
| ---
| ---
|TODO
|}

== References ==


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

TMBR: May 2026

2026-05-06T05:42:44Z

RobinCodes: /* Holdouts */ Added BB(2,6) progress

{{TMBRnav|April 2026|June 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).''

== BB Adjacent ==
[[General Recursive Function|General Recursive Functions]]:

* Some new cryptids were hand-built:
** Size 56, by Shawn on 2 May (simulating 5x+1 problem starting at 7).[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Size 49, by aparker, star and Shawn on 3 May (simulating [[wikipedia:Brocard's_problem|Brocard's problem]]).[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf]
* The first non-trivial divergent GRF was found (size 15). It halts iff there exists some n ≥ 1 such that n+3 divides <math>Tetr(n) = \frac{n(n+1)(n+2)}{6}</math>.[https://discord.com/channels/960643023006490684/960643023530762341/1500584497542987776] aparker[https://discord.com/channels/960643023006490684/960643023530762341/1500587569514283098] and star[https://discord.com/channels/960643023006490684/960643023530762341/1500595210919346337] proved that there is no such n.

== Holdouts ==

*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 536,112 to '''527,232''' via Enumerate.py and TM-enum, a '''1.66%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1500218448951775383][https://discord.com/channels/960643023006490684/1084047886494470185/1501436678823477368 <nowiki>[7]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 20K more subtasks, increasing the number of holdouts to '''749,156,843'''. A total of 24K subtasks out of the 1 million subtasks (or '''24%''') have been enumerated.[https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[8]</nowiki>]

[[Category:This Month in Beaver Research|2026-05]]

TMBR: April 2026

2026-05-03T15:22:04Z

RobinCodes: Added more explanation

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== 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]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a size 29 program that is tetrational (depending on what we consider tetrational), see [https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330 Discord].
**Racheline also created <math>f_\omega</math> programs starting from size 86, see [https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789 Discord].
**Finally, Racheline created <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]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>]
* [[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]. (Altough 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].

== 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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 225 respectively, due to the recent equivalence reductions (10 machines total).
*[[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>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine known to halt from 2024 June (but not simulated there by a direct simulator), where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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

2026-05-03T15:01:06Z

RobinCodes: /* BB Adjacent */ Add -d's contribution to equivalences

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== 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]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a size 29 program that is tetrational (depending on what we consider tetrational), see [https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330 Discord].
**Racheline also created <math>f_\omega</math> programs starting from size 86, see [https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789 Discord].
**Finally, Racheline created <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]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>]
* [[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]. (Altough 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].

== 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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 225 respectively, due to the recent equivalence reductions (10 machines total).
*[[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>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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

2026-05-03T14:56:33Z

RobinCodes: /* BB Adjacent */ Updated JBB

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== 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]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a size 29 program that is tetrational (depending on what we consider tetrational), see [https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330 Discord].
**Racheline also created <math>f_\omega</math> programs starting from size 86, see [https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789 Discord].
**Finally, Racheline created <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]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>]
* [[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]. (Altough 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].

== 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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 225 respectively, due to the recent equivalence reductions (10 machines total).
*[[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>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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

2026-05-03T13:47:07Z

RobinCodes: /* Holdouts */ Remove completed TODO

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== 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]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a size 29 program that is tetrational (depending on what we consider tetrational), see [https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330 Discord].
**Racheline also created <math>f_\omega</math> programs starting from size 86, see [https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789 Discord].
**Finally, Racheline created <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]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>]
* [[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]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== 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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 225 respectively, due to the recent equivalence reductions (10 machines total).
*[[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>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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

2026-05-03T13:46:28Z

RobinCodes: /* Holdouts */ Add mxdys BB6 progress graph 2026

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== 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]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a size 29 program that is tetrational (depending on what we consider tetrational), see [https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330 Discord].
**Racheline also created <math>f_\omega</math> programs starting from size 86, see [https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789 Discord].
**Finally, Racheline created <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]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>]
* [[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]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== 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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 225 respectively, due to the recent equivalence reductions (10 machines total).
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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]]

File:BB6 progress Q1 2026.png

2026-05-03T13:45:23Z

RobinCodes:

BB6 progress in 2026 graph by mxdys

User:RobinCodes/Work on BB Domains

2026-05-03T13:41:24Z

RobinCodes: Added BB(6) progress

== Progress ==
As of May 2026,

'''Domains with active filtering going on (decider-wise): [[BB(2,6)]], [[BB(7)]] - Andrew Ducharme'''

{| class="wikitable"
|+ Work on BB Domains
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
!8-state
!9-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
2 Jul 2024
| style="background: #A8613E;" | 1104
[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Holdouts list]

(2 informal)

16+6 Cryptids

[[BB(6)]]
| style="background: #A06B10;" | 17,823,260
No Handwork
Andrew Ducharme, Terry Ligocki
Phase 2, Stage 6

[[BB(7)]]
| style="background: #AA8519;" |Expected no. of holdouts: 147 billion (<math>1.47 \cdot 10^{11}</math>)
[[BB(8)]]
| style="background: #ECC652;" |
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: #A8613E;" | 4
[[BB(3,3)#Holdouts|Holdouts list]]

1 Cryptid, Exhausted

[[BB(3,3)]]
| style="background: #A06B10;" | 5,641,006
No Handwork

Terry Ligocki, Andrew Ducharme Phase 2, Stage 3

[[BB(4,3)]]
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #A8973E;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #A06B10;" | 12,049,358
No Handwork
[[User:XnoobSpeakable|XnoobSpeakable,]] [[User:WarpedWartWars|Lúkos]], Andrew Ducharme

Phase 3

[[BB(3,4)]]
| style="background: #D8B446;" |Expected no. of holdouts: 1.53 quadrillion (<math>1.53 \cdot 10^{15}</math>)
[[BB(4,4)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
! 5-symbol
| style="background: #A8613E;" | 60
[[BB(2,5)#Holdouts|Holdouts list]]
(6 informal)

2+4 Cryptids

[[BB(2,5)]]
| style="background: #D8B446;" |Lower bound: (due Racheline 2024)<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
Expected no. of holdouts: 11.3 trillion (<math>1.13 \cdot 10^{13}</math>)
[[BB(3,5)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
! 6-symbol
| style="background: #A06B10;" | 533,764
No Handwork
Andrew Ducharme,
Terry Ligocki

Phase 2 Stage 4

[[BB(2,6)]]
| style="background: #ECC652;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
!7-symbol
| style="background: #AA8519;" |Enumeration started!
24% Done

Expected no. of holdouts: 3 billion (<math>3 \cdot 10^9</math>)

Current: 750 million

[[BB(2,7)]]
| style="background: #ECC652;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
!8-symbol
| style="background: #ECC652;" |Expected no. of holdouts: 5.51 trillion (<math display="inline">5.51 \cdot 10^{12}</math>)
[[BB(2,8)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|}
Futher domains with some information:

* [[BB(11)]]: Lower bound: (Racheline 2024)
<math display="block">f_\omega^2(2 \uparrow\uparrow 12)</math>

* [[BB(12)]]: Lower bound: (Racheline 2024)
<math display="block">f_\omega^4(2 \uparrow\uparrow\uparrow 4-3)</math>
== Enumeration ==
The enumeration process for [[BB(2,7)]] has started, and 10% has so far been completed. [[BB(8)]] is (tested from a random sample) even harder to enumerate. These processes will take a very long time and much collaborative work to complete. As for BB(3x5), BB(4x4), BB(5x3), enumeration may still be possible, but also incredibly difficult. Any domains after these are probably complete out-of-reach of current technology, just to enumerate. Using crowdsourced enumeration, one more layer may be possible.

== Informal Proofs ==
'''Unconfirmed, informal proofs of machines for each domain by equivalence class: 8 total'''
{| class="wikitable"
!Domain
!2-state
!3-state
!4-state
!5-state
!6-state
|-
!2-symbol
|0
|0
|0
|0
|'''2'''
|-
!3-symbol
|0
|0
| ----
| ---
| ---
|-
!4-symbol
|0
| ---
| ---
| ---
| ---
|-
!5-symbol
|'''6'''
| ---
| ---
| ---
| ---
|}
=== [[BB(6)]]: 1 March 2026 ===

===== 2 informal =====

*mxdys, Pomme, Autumn Pan, vyx[https://discord.com/channels/960643023006490684/1239205785913790465/1441124403801755730 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1443295684878143579 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1444047803935490190 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1444051070186819836 <nowiki>[4]</nowiki>]: <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code>
*<code>1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA</code> (halt, Racheline, informal language)

=== [[BB(2,5)]]: 1 April 2026 ===

===== 5+1 informal =====

* Dyuan:
*# Dekaheptoid, unverified: [[1RB3RB1LB---2RB_2LA1RA4LB2LA2RA]] ([https://discord.com/channels/960643023006490684/1259770421046411285/1267650177389432913 source]), equivalently (1.2) <code>1RB3RA2LB1LB1RB_2LA2RA4LA1LA---</code>
* Legion:
*# <code>1RB2RA3LA4LA2RB_2LA---1LA1RA3RA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 1])
*# <code>1RB3LA4LA1LA2RA_2LA4RB---0RA0LA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 2])
*# <code>1RB3LA4LA2RB1LA_2LA4RB---3RA3LA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 3])
* [https://discord.com/channels/960643023006490684/1259770421046411285/1329663999700111471 Racheline]: <code>1RB2LB---4LB0RB_1LA3RB4RB4RA1LB</code>
=== [[BB(3,3)]]: 28 March 2026 ===
''None''

TMBR: April 2026

2026-05-03T13:39:56Z

RobinCodes: /* Holdouts */ Fix simulation progress

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== 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]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a size 29 program that is tetrational (depending on what we consider tetrational), see [https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330 Discord].
**Racheline also created <math>f_\omega</math> programs starting from size 86, see [https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789 Discord].
**Finally, Racheline created <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]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>]
* [[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]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== 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%
|}

*[[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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 225 respectively, due to the recent equivalence reductions (10 machines total).
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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

2026-05-03T13:34:43Z

RobinCodes: /* Holdouts */ Simulation results

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== 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]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a size 29 program that is tetrational (depending on what we consider tetrational), see [https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330 Discord].
**Racheline also created <math>f_\omega</math> programs starting from size 86, see [https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789 Discord].
**Finally, Racheline created <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]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>]
* [[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]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== 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%
|}

*[[BB(6)]]: Reduction: 57. No. of TMs to simulate to 1e14: 161 (reduction: 10). To 1e15: 226 (reduction: 12).
**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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is 161 & 226 respectively, due to the recent equivalence reductions (10 machines total). TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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

2026-05-03T13:07:29Z

RobinCodes: /* Holdouts */ Added Racheline's Fractran programs

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== 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]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
**Racheline created a size 29 program that is tetrational (depending on what we consider tetrational), see [https://discord.com/channels/960643023006490684/1438019511155691521/1489361701727109330 Discord].
**Racheline also created <math>f_\omega</math>programs starting from size 86, see [https://discord.com/channels/960643023006490684/1438019511155691521/1489473702000201789 Discord].
**Finally, Racheline created <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]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BB\(38) has been solved] (BB\(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]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== 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%
|}

*[[BB(6)]]
**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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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

2026-05-03T12:53:26Z

RobinCodes: /* BB Adjacent */ Better dimensions for image

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== 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]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
* [[General Recursive Function]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BB\(38) has been solved] (BB\(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]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== 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%
|}

*[[BB(6)]]
**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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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

2026-05-03T12:52:40Z

RobinCodes: Remove "Misc" section as topic was about Fractran (migrated to BB Adj)

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir-family]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
* [[General Recursive Function]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BB\(38) has been solved] (BB\(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]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== 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%
|}

*[[BB(6)]]
**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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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

2026-05-03T12:52:06Z

RobinCodes: /* BB Adjacent */ Added Fractran space-time diagram

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==
[[File:Space Needle.webp|alt=Space-time diagram of Space Needle in Fractran.|thumb|Space-time diagram of Space Needle in Fractran.]]
* [[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir-family]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
* [[General Recursive Function]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BB\(38) has been solved] (BB\(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]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Misc ==
* Katelyn Doucette completed [https://github.com/Laturas/FractranVisualizer a visualizer] for Fractran space-time diagrams.[https://discord.com/channels/960643023006490684/1438019511155691521/1488727841951449088 <nowiki>[9]</nowiki>]

== 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%
|}

*[[BB(6)]]
**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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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

2026-05-03T12:49:39Z

RobinCodes: Fix references

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==

* [[General Recursive Function]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677 <nowiki>[3]</nowiki>]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482 <nowiki>[4]</nowiki>]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125 <nowiki>[5]</nowiki>][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863 <nowiki>[6]</nowiki>]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788 <nowiki>[7]</nowiki>]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189 <nowiki>[8]</nowiki>]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BB\(38) has been solved] (BB\(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]
*[[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir-family]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Misc ==
* Katelyn Doucette completed [https://github.com/Laturas/FractranVisualizer a visualizer] for Fractran space-time diagrams.[https://discord.com/channels/960643023006490684/1438019511155691521/1488727841951449088 <nowiki>[9]</nowiki>]

== 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%
|}

*[[BB(6)]]
**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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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

2026-05-03T12:42:37Z

RobinCodes: Add Fractran and introductory text

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for [[Fractran]] space-time diagrams. BBf(22) has been solved except for the [[Fractran#Fenrir|Fenrir-family]][https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 <nowiki>[1]</nowiki>], enumeration of BBf(23) will take roughly 10 days.[https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp <nowiki>[2]</nowiki>] There was a 40% reduction in [[BB(4,3)]], and we also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==

* [[General Recursive Function]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189]
* [[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BB\(38) has been solved] (BB\(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]
*[[Fractran]]
**[https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 BBf(22) was solved] with the exception of the [[Fractran#Fenrir|Fenrir-family]].
**Katelyn Doucette [https://github.com/Laturas/FractranVisualizer created a visualizer for Fractran space-time diagrams].
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].

== Misc ==
* Katelyn Doucette completed [https://github.com/Laturas/FractranVisualizer a visualizer] for Fractran space-time diagrams.[https://discord.com/channels/960643023006490684/1438019511155691521/1488727841951449088 <nowiki>[7]</nowiki>]

== 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%
|}

*[[BB(6)]]
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[9]</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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>[10]</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>[11]</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>[12]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[13]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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>[14]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[15]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[16]</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>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[18]</nowiki>]

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

Fractran

2026-05-03T12:28:33Z

RobinCodes: /* BBf(21) */ Formatting

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

BBf(21) and below are solved. BBf(22) is the smallest domain to contain a Cryptid, all other machines are solved there. Enumeration of BBf(23) will take roughly 10 days ([https://github.com/int-y1/BBFractran/blob/main/enumerate/fractran20260416.cpp Enum script]).

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

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

Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1447069110541484146 7] and [https://discord.com/channels/960643023006490684/1438019511155691521/1453213088630444168 24 Dec 2025]
6 Holdouts: Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1452913055053778945 23 Dec 2025]
|-
|21
|31,957,632
|<code>[7/15, 4/3, 27/14, 5/2, 9/5]</code>
|<math>\begin{bmatrix}
0 & -1 & -1 & 1 \\
2 & -1 & 0 & 0 \\
-1 & 3 & 0 & -1 \\
-1 & 0 & 1 & 0 \\
0 & 2 & -1 & 0
\end{bmatrix}</math>
|Jason Yuen (@-d) [https://discord.com/channels/960643023006490684/1438019511155691521/1439759182587891894 16 Nov 2025]
|140 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 25 Jan 2026]
Claude Opus 4.6 proof of nonhalting of all 140: [https://discord.com/channels/960643023006490684/1438019511155691521/1485168251997786173 28 March 2026]
|-
|22
|<math>> 1.146 \times 10^{62}</math>
|<code>[1/12, 9/10, 14/3, 11/2, 5/7, 3/11]</code>
|<math>\begin{bmatrix}
-2 & -1 & 0 & 0 & 0 \\
-1 & 2 & -1 & 0 & 0 \\
1 & -1 & 0 & 1 & 0 \\
-1 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & -1 & 0 \\
0 & 1 & 0 & 0 & -1
\end{bmatrix}</math>
|Shawn Ligocki (@sligocki) [https://discord.com/channels/960643023006490684/1438019511155691521/1448912286713384961 11 Dec 2025] and Jason Yuen (@-d)[https://discord.com/channels/960643023006490684/1438019511155691521/1448953682237460480 <nowiki>[1]</nowiki>]
|2003 holdouts remain. [https://discord.com/channels/960643023006490684/1438019511155691521/1464873923647639703 25 Jan 2026]
Claude Opus 4.6 gave a proof of all machines but the Fenrir-family, see [https://discord.com/channels/960643023006490684/1438019511155691521/1493027835559022824 Discord].

Known [[Cryptid|Cryptids]]:

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

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

=== Behavior of Champions ===

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

This family obeys the following rules:

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

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

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

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

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

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

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

which follows the reasonably "lucky" trajectory:

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

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

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

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

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

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

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

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

== Cryptids ==

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Start: A(7, 1)

A(1, b) --> halt

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

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

== References ==
<references />

[[Category:Functions]]

Fractran

2026-05-03T12:27:34Z

RobinCodes: /* Champions */ Added paragraph about domains and BBf(22) tentatively proved

TMBR: May 2026

2026-05-03T11:50:23Z

RobinCodes: /* Holdouts */ Preassume reference

{{TMBRnav|April 2026|June 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).''

== Holdouts ==

*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 536,112 to '''533,764''' via Enumerate.py, a '''0.44%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1500218448951775383]
*[[BB(2,7)]]
** Terry Ligocki enumerated 20K more subtasks, increasing the number of holdouts to '''749,156,843'''. A total of 24K subtasks out of the 1 million subtasks (or '''24%''') have been enumerated.[https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659 <nowiki>[2]</nowiki>]

[[Category:This Month in Beaver Research|2026-05]]

TMBR: April 2026

2026-05-03T11:49:52Z

RobinCodes: /* Holdouts */ Rollback BB(2,7) updates since it happened in May

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for Fractran space-time diagrams. We also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==

* [[General Recursive Function]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].
*[[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BB\(38) has been solved] (BB\(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]

== Misc ==
* Katelyn Doucette completed [https://github.com/Laturas/FractranVisualizer a visualizer] for Fractran space-time diagrams.[https://discord.com/channels/960643023006490684/1438019511155691521/1488727841951449088 <nowiki>[7]</nowiki>]

== 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%
|}

*[[BB(6)]]
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[9]</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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>[10]</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>[11]</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>[12]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[13]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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>[14]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[15]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[16]</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>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[18]</nowiki>]

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

TMBR: April 2026

2026-05-03T11:48:19Z

RobinCodes: /* Holdouts */ Roll back BB(2,6) update since it happened in May

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for Fractran space-time diagrams. We also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==

* [[General Recursive Function]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].
*[[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BB\(38) has been solved] (BB\(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]

== Misc ==
* Katelyn Doucette completed [https://github.com/Laturas/FractranVisualizer a visualizer] for Fractran space-time diagrams.[https://discord.com/channels/960643023006490684/1438019511155691521/1488727841951449088 <nowiki>[7]</nowiki>]

== 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%
|}

*[[BB(6)]]
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[9]</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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>[10]</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>[11]</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>[12]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[13]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[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>[14]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[15]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[16]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 140K more subtasks, increasing the number of holdouts to '''749,156,843'''. A total of 240K subtasks out of the 1 million subtasks (or '''24%''') 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>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[18]</nowiki>]

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

User:RobinCodes/Work on BB Domains

2026-05-03T11:43:01Z

RobinCodes: /* Progress */ Added BB(3,4) & (4,3) progress

== Progress ==
As of May 2026,

'''Domains with active filtering going on (decider-wise): [[BB(2,6)]], [[BB(7)]] - Andrew Ducharme'''

{| class="wikitable"
|+ Work on BB Domains
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
!8-state
!9-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
2 Jul 2024
| style="background: #A8613E;" | 1161
[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Holdouts list]

(2 informal)

16+6 Cryptids

[[BB(6)]]
| style="background: #A06B10;" | 17,823,260
No Handwork
Andrew Ducharme, Terry Ligocki
Phase 2, Stage 6

[[BB(7)]]
| style="background: #AA8519;" |Expected no. of holdouts: 147 billion (<math>1.47 \cdot 10^{11}</math>)
[[BB(8)]]
| style="background: #ECC652;" |
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: #A8613E;" | 4
[[BB(3,3)#Holdouts|Holdouts list]]

1 Cryptid, Exhausted

[[BB(3,3)]]
| style="background: #A06B10;" | 5,641,006
No Handwork

Terry Ligocki, Andrew Ducharme Phase 2, Stage 3

[[BB(4,3)]]
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #A8973E;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #A06B10;" | 12,049,358
No Handwork
[[User:XnoobSpeakable|XnoobSpeakable,]] [[User:WarpedWartWars|Lúkos]], Andrew Ducharme

Phase 3

[[BB(3,4)]]
| style="background: #D8B446;" |Expected no. of holdouts: 1.53 quadrillion (<math>1.53 \cdot 10^{15}</math>)
[[BB(4,4)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
! 5-symbol
| style="background: #A8613E;" | 60
[[BB(2,5)#Holdouts|Holdouts list]]
(6 informal)

2+4 Cryptids

[[BB(2,5)]]
| style="background: #D8B446;" |Lower bound: (due Racheline 2024)<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
Expected no. of holdouts: 11.3 trillion (<math>1.13 \cdot 10^{13}</math>)
[[BB(3,5)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
! 6-symbol
| style="background: #A06B10;" | 533,764
No Handwork
Andrew Ducharme,
Terry Ligocki

Phase 2 Stage 4

[[BB(2,6)]]
| style="background: #ECC652;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
!7-symbol
| style="background: #AA8519;" |Enumeration started!
24% Done

Expected no. of holdouts: 3 billion (<math>3 \cdot 10^9</math>)

Current: 750 million

[[BB(2,7)]]
| style="background: #ECC652;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
!8-symbol
| style="background: #ECC652;" |Expected no. of holdouts: 5.51 trillion (<math display="inline">5.51 \cdot 10^{12}</math>)
[[BB(2,8)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|}
Futher domains with some information:

* [[BB(11)]]: Lower bound: (Racheline 2024)
<math display="block">f_\omega^2(2 \uparrow\uparrow 12)</math>

* [[BB(12)]]: Lower bound: (Racheline 2024)
<math display="block">f_\omega^4(2 \uparrow\uparrow\uparrow 4-3)</math>
== Enumeration ==
The enumeration process for [[BB(2,7)]] has started, and 10% has so far been completed. [[BB(8)]] is (tested from a random sample) even harder to enumerate. These processes will take a very long time and much collaborative work to complete. As for BB(3x5), BB(4x4), BB(5x3), enumeration may still be possible, but also incredibly difficult. Any domains after these are probably complete out-of-reach of current technology, just to enumerate. Using crowdsourced enumeration, one more layer may be possible.

== Informal Proofs ==
'''Unconfirmed, informal proofs of machines for each domain by equivalence class: 8 total'''
{| class="wikitable"
!Domain
!2-state
!3-state
!4-state
!5-state
!6-state
|-
!2-symbol
|0
|0
|0
|0
|'''2'''
|-
!3-symbol
|0
|0
| ----
| ---
| ---
|-
!4-symbol
|0
| ---
| ---
| ---
| ---
|-
!5-symbol
|'''6'''
| ---
| ---
| ---
| ---
|}
=== [[BB(6)]]: 1 March 2026 ===

===== 2 informal =====

*mxdys, Pomme, Autumn Pan, vyx[https://discord.com/channels/960643023006490684/1239205785913790465/1441124403801755730 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1443295684878143579 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1444047803935490190 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1444051070186819836 <nowiki>[4]</nowiki>]: <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code>
*<code>1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA</code> (halt, Racheline, informal language)

=== [[BB(2,5)]]: 1 April 2026 ===

===== 5+1 informal =====

* Dyuan:
*# Dekaheptoid, unverified: [[1RB3RB1LB---2RB_2LA1RA4LB2LA2RA]] ([https://discord.com/channels/960643023006490684/1259770421046411285/1267650177389432913 source]), equivalently (1.2) <code>1RB3RA2LB1LB1RB_2LA2RA4LA1LA---</code>
* Legion:
*# <code>1RB2RA3LA4LA2RB_2LA---1LA1RA3RA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 1])
*# <code>1RB3LA4LA1LA2RA_2LA4RB---0RA0LA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 2])
*# <code>1RB3LA4LA2RB1LA_2LA4RB---3RA3LA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 3])
* [https://discord.com/channels/960643023006490684/1259770421046411285/1329663999700111471 Racheline]: <code>1RB2LB---4LB0RB_1LA3RB4RB4RA1LB</code>
=== [[BB(3,3)]]: 28 March 2026 ===
''None''

TMBR: April 2026

2026-05-03T11:40:56Z

RobinCodes: /* Holdouts */ Fix references

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for Fractran space-time diagrams. We also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==

* [[General Recursive Function]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].
*[[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BB\(38) has been solved] (BB\(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]

== Misc ==
* Katelyn Doucette completed [https://github.com/Laturas/FractranVisualizer a visualizer] for Fractran space-time diagrams.[https://discord.com/channels/960643023006490684/1438019511155691521/1488727841951449088 <nowiki>[7]</nowiki>]

== 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
|533,764
|11,241
|2.06%
|}

*[[BB(6)]]
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[9]</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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>[10]</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>[11]</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>[12]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[13]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''533,764''' via Enumerate.py, a '''2.06%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[15]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[16]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1500218448951775383 <nowiki>[17]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 140K more subtasks, increasing the number of holdouts to '''749,156,843'''. A total of 240K subtasks out of the 1 million subtasks (or '''24%''') 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>[18]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[19]</nowiki>]

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

TMBR: April 2026

2026-05-03T11:34:27Z

RobinCodes: Add BB\

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for Fractran space-time diagrams. We also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==

* [[General Recursive Function]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].
*[[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]]
**[https://discord.com/channels/960643023006490684/1355653587824283678/1492950712940892210 BB\(38) has been solved] (BB\(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]

== Misc ==
* Katelyn Doucette completed [https://github.com/Laturas/FractranVisualizer a visualizer] for Fractran space-time diagrams.[https://discord.com/channels/960643023006490684/1438019511155691521/1488727841951449088 <nowiki>[7]</nowiki>]

== 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
|533,764
|11,241
|2.06%
|}

*[[BB(6)]]
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[9]</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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>[10]</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>TODO</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>[11]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[12]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''533,764''' via Enumerate.py, a '''2.06%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[13]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[15]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1500218448951775383 <nowiki>[16]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 140K more subtasks, increasing the number of holdouts to '''749,156,843'''. A total of 240K subtasks out of the 1 million subtasks (or '''24%''') 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>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[18]</nowiki>]

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

Lambda Calculus

2026-05-03T11:34:19Z

RobinCodes: /* Champions */ Add smallest known Cryptid

'''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 ==

* https://oeis.org/A333479
* [https://tromp.github.io/blog/2026/01/28/largest-number-revised The largest number representable in 64 bits]. 28 Jan 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]]

TMBR: April 2026

2026-05-03T11:07:01Z

RobinCodes: Note about 1 solved machine in first reduction this month

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for Fractran space-time diagrams. We also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==

* [[General Recursive Function]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].
*TODO: BB\ ([[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]])

== Misc ==
* Katelyn Doucette completed [https://github.com/Laturas/FractranVisualizer a visualizer] for Fractran space-time diagrams.[https://discord.com/channels/960643023006490684/1438019511155691521/1488727841951449088 <nowiki>[7]</nowiki>]

== 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
|533,764
|11,241
|2.06%
|}

*[[BB(6)]]
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[9]</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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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 new equivalences. This means there is now only 1 holdout considered "informal", which is actually very formal, but depends on Baker's theorem (actually, more restricted than that is enough, see above), and therefore 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.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>[10]</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>TODO</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>[11]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[12]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''533,764''' via Enumerate.py, a '''2.06%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[13]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[15]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1500218448951775383 <nowiki>[16]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 140K more subtasks, increasing the number of holdouts to '''749,156,843'''. A total of 240K subtasks out of the 1 million subtasks (or '''24%''') 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>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[18]</nowiki>]

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

TMBR: April 2026

2026-05-03T11:01:43Z

RobinCodes: Added formalisation of Baker-Wüsthold machine

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for Fractran space-time diagrams. We also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==

* [[General Recursive Function]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].
*TODO: BB\ ([[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]])

== Misc ==
* Katelyn Doucette completed [https://github.com/Laturas/FractranVisualizer a visualizer] for Fractran space-time diagrams.[https://discord.com/channels/960643023006490684/1438019511155691521/1488727841951449088 <nowiki>[7]</nowiki>]

== 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
|533,764
|11,241
|2.06%
|}

*[[BB(6)]]
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[9]</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 result in Number Theory, which has not yet been formalised in any formal language, and doing so would be a large project. 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.
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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]) came from new equivalences.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>[10]</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>TODO</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>[11]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[12]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''533,764''' via Enumerate.py, a '''2.06%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[13]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[15]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1500218448951775383 <nowiki>[16]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 140K more subtasks, increasing the number of holdouts to '''749,156,843'''. A total of 240K subtasks out of the 1 million subtasks (or '''24%''') 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>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[18]</nowiki>]

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

BB(2,6)

2026-05-03T10:53:35Z

RobinCodes: Update 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|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="29" |[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
|-
|536,112
|533,764
|0.43%
|8778.5
| ---
| ---
|TODO: [https://discord.com/channels/960643023006490684/1084047886494470185/1500218448951775383 Discord]
|}

== References ==


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

TMBR: April 2026

2026-05-03T10:49:19Z

RobinCodes: Updated BB(2,7)

{{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 [[BMO]]. Two informally proven machines were formalised into Rocq in [[BB(2,5)]], and Katelyn Doucette created a visualizer for Fractran space-time diagrams. We also shot below 18 million holdouts for [[BB(7)]].

== BB Adjacent ==

* [[General Recursive Function]]
** 3 Apr: Jacob Mandelson proved the values up to BBµ(7).[https://discord.com/channels/960643023006490684/1447627603698647303/1489782558446321677]
** 8 Apr: Jacob constructed a size 141 [[Cryptid]].[https://discord.com/channels/960643023006490684/1447627603698647303/1491642156295913482]
** 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1492990073820545125][https://discord.com/channels/960643023006490684/1447627603698647303/1493060638896033863]
** 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.[https://discord.com/channels/960643023006490684/1447627603698647303/1494396445208608788]
** 29 Apr: Shawn Ligocki found a new BBµ(14) champion using the min operator.[https://discord.com/channels/960643023006490684/1447627603698647303/1499137558695641189]
*[https://discord.com/channels/960643023006490684/1362008236118511758/1493973516326928494 "BB" for Sokoban has been shared on the Discord server]. (Altough it is computable like [[Bug Game]], so we wouldn't call it a BB-function.)
*Jumping Busy Beaver has been introduced, JBB(2,2,0) is known along with some lower bounds on small domains, see the [https://discord.com/channels/960643023006490684/1496202019206336664/1496202019206336664 Discord thread].
*TODO: BB\ ([[Busy Beaver for lambda calculus|Busy Beaver for Lambda Calculus]])

== Misc ==
* Katelyn Doucette completed [https://github.com/Laturas/FractranVisualizer a visualizer] for Fractran space-time diagrams.[https://discord.com/channels/960643023006490684/1438019511155691521/1488727841951449088 <nowiki>[7]</nowiki>]

== 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
|533,764
|11,241
|2.06%
|}

*[[BB(6)]]
**Discord user sheep discovered[https://discord.com/channels/960643023006490684/1448375857046360094/1490939334092787722 <nowiki>[8]</nowiki>][https://discord.com/channels/960643023006490684/1448375857046360094/1490772706269069313 <nowiki>[9]</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}}
**Alistaire [https://discord.com/channels/960643023006490684/1477591686514212894/1490470766116864291 simulated a machine] to 1e15.
**Discord user The_Real_Fourious_Banana [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 simulated another TM] to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
**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]) came from new equivalences.
**Later, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1499000732236382358 released] a new holdouts list of '''1104''' machines where more equivalence classes have been merged.
**Along with [https://discord.com/channels/960643023006490684/1477591686514212894/1495412160237539338 the 1 TM simulated by Discord user @furiousbanana] ([https://discord.com/channels/960643023006490684/1477591686514212894/1499712071946862655 Link] to further simulation), the number of machines to simulate to 1e14 & 1e15 is X & Y respectively, due to the recent equivalence reductions. TODO: Add
**TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
*[[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>[10]</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>TODO</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>[11]</nowiki>][https://discord.com/channels/960643023006490684/1259770421046411285/1491830661512958185 <nowiki>[12]</nowiki>] which confirmed the already existing moderately formal argument further. {{TM|1RB3LA4LA2RB1LA_2LA4RB---3RA3LA|halt}} is the only remaining machine suspected to halt from 2024 June, where the other two machines were first found to halt (see [https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 Discord]).
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 545,005 to '''533,764''' via Enumerate.py, a '''2.06%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 <nowiki>[13]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1495650803967463464 <nowiki>[14]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1497280483275575347 <nowiki>[15]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1500218448951775383 <nowiki>[16]</nowiki>]
*[[BB(2,7)]]
** Terry Ligocki enumerated 140K more subtasks, increasing the number of holdouts to '''749,156,843'''. A total of 240K subtasks out of the 1 million subtasks (or '''24%''') 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>[17]</nowiki>][https://discord.com/channels/960643023006490684/1084047886494470185/1498198584208658443 <nowiki>[18]</nowiki>]

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

User:RobinCodes/Work on BB Domains

2026-05-03T10:46:34Z

RobinCodes: /* Progress */ Update BB(2,6) and BB(2,7)

== Progress ==
As of May 2026,

'''Domains with active filtering going on (decider-wise): [[BB(2,6)]], [[BB(7)]] - Andrew Ducharme'''

{| class="wikitable"
|+ Work on BB Domains
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
!8-state
!9-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
2 Jul 2024
| style="background: #A8613E;" | 1161
[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Holdouts list]

(2 informal)

16+6 Cryptids

[[BB(6)]]
| style="background: #A06B10;" | 17,823,260
No Handwork
Andrew Ducharme, Terry Ligocki
Phase 2, Stage 6

[[BB(7)]]
| style="background: #AA8519;" |Expected no. of holdouts: 147 billion (<math>1.47 \cdot 10^{11}</math>)
[[BB(8)]]
| style="background: #ECC652;" |
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: #A8613E;" | 4
[[BB(3,3)#Holdouts|Holdouts list]]

1 Cryptid, Exhausted

[[BB(3,3)]]
| style="background: #A06B10;" | 9,401,447
No Handwork

Terry Ligocki Phase 2, Stage 2

[[BB(4,3)]]
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #ECC652;" |
| style="background: #A8973E;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #A06B10;" | 12,435,284
No Handwork

[[User:XnoobSpeakable|XnoobSpeakable,]] [[User:WarpedWartWars|Lúkos]]

Phase 2, Stage 10C

[[BB(3,4)]]
| style="background: #D8B446;" |Expected no. of holdouts: 1.53 quadrillion (<math>1.53 \cdot 10^{15}</math>)
[[BB(4,4)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
! 5-symbol
| style="background: #A8613E;" | 60
[[BB(2,5)#Holdouts|Holdouts list]]
(6 informal)

2+4 Cryptids

[[BB(2,5)]]
| style="background: #D8B446;" |Lower bound: (due Racheline 2024)<math>> f_\omega(2 \uparrow^{15} 5) > f_\omega^2(15)</math>
Expected no. of holdouts: 11.3 trillion (<math>1.13 \cdot 10^{13}</math>)
[[BB(3,5)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
! 6-symbol
| style="background: #A06B10;" | 533,764
No Handwork
Andrew Ducharme,
Terry Ligocki

Phase 2 Stage 4

[[BB(2,6)]]
| style="background: #ECC652;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
!7-symbol
| style="background: #AA8519;" |Enumeration started!
24% Done

Expected no. of holdouts: 3 billion (<math>3 \cdot 10^9</math>)

Current: 750 million

[[BB(2,7)]]
| style="background: #ECC652;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|-
!8-symbol
| style="background: #ECC652;" |Expected no. of holdouts: 5.51 trillion (<math display="inline">5.51 \cdot 10^{12}</math>)
[[BB(2,8)]]
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
| style="background: #A8973E;" |
|}
Futher domains with some information:

* [[BB(11)]]: Lower bound: (Racheline 2024)
<math display="block">f_\omega^2(2 \uparrow\uparrow 12)</math>

* [[BB(12)]]: Lower bound: (Racheline 2024)
<math display="block">f_\omega^4(2 \uparrow\uparrow\uparrow 4-3)</math>
== Enumeration ==
The enumeration process for [[BB(2,7)]] has started, and 10% has so far been completed. [[BB(8)]] is (tested from a random sample) even harder to enumerate. These processes will take a very long time and much collaborative work to complete. As for BB(3x5), BB(4x4), BB(5x3), enumeration may still be possible, but also incredibly difficult. Any domains after these are probably complete out-of-reach of current technology, just to enumerate. Using crowdsourced enumeration, one more layer may be possible.

== Informal Proofs ==
'''Unconfirmed, informal proofs of machines for each domain by equivalence class: 8 total'''
{| class="wikitable"
!Domain
!2-state
!3-state
!4-state
!5-state
!6-state
|-
!2-symbol
|0
|0
|0
|0
|'''2'''
|-
!3-symbol
|0
|0
| ----
| ---
| ---
|-
!4-symbol
|0
| ---
| ---
| ---
| ---
|-
!5-symbol
|'''6'''
| ---
| ---
| ---
| ---
|}
=== [[BB(6)]]: 1 March 2026 ===

===== 2 informal =====

*mxdys, Pomme, Autumn Pan, vyx[https://discord.com/channels/960643023006490684/1239205785913790465/1441124403801755730 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1443295684878143579 <nowiki>[2]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1444047803935490190 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1443295684878143579/1444051070186819836 <nowiki>[4]</nowiki>]: <code>1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---</code>
*<code>1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA</code> (halt, Racheline, informal language)

=== [[BB(2,5)]]: 1 April 2026 ===

===== 5+1 informal =====

* Dyuan:
*# Dekaheptoid, unverified: [[1RB3RB1LB---2RB_2LA1RA4LB2LA2RA]] ([https://discord.com/channels/960643023006490684/1259770421046411285/1267650177389432913 source]), equivalently (1.2) <code>1RB3RA2LB1LB1RB_2LA2RA4LA1LA---</code>
* Legion:
*# <code>1RB2RA3LA4LA2RB_2LA---1LA1RA3RA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 1])
*# <code>1RB3LA4LA1LA2RA_2LA4RB---0RA0LA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 2])
*# <code>1RB3LA4LA2RB1LA_2LA4RB---3RA3LA</code> ([https://discord.com/channels/960643023006490684/1084047886494470185/1254518334406266964 3])
* [https://discord.com/channels/960643023006490684/1259770421046411285/1329663999700111471 Racheline]: <code>1RB2LB---4LB0RB_1LA3RB4RB4RA1LB</code>
=== [[BB(3,3)]]: 28 March 2026 ===
''None''

TMBR: April 2026

2026-05-03T10:43:29Z

RobinCodes: Fixed BB(2,6) holdout count and reduction %

TMBR: April 2026

2026-05-03T10:42:20Z

RobinCodes: added BB(2,6) progress, fixed references

TMBR: April 2026

2026-05-03T10:35:21Z

RobinCodes: /* Holdouts */ Added more about BB(2,5)

TMBR: April 2026

2026-05-03T10:07:38Z

RobinCodes: /* BB Adjacent */ Add JBB

TMBR: April 2026

2026-05-03T09:58:12Z

RobinCodes: /* BB Adjacent */ Note about Sokoban being computable

TMBR: April 2026

2026-05-03T09:57:09Z

RobinCodes: /* BB Adjacent */ Added Sokoban BB

TMBR: April 2026

2026-05-03T09:27:09Z

RobinCodes: More simulation by @furiousbanana

TMBR: April 2026

2026-05-03T09:25:57Z

RobinCodes: /* Holdouts */ TODO simulation

User:RobinCodes/Valuable user pages

2026-04-23T18:21:36Z

RobinCodes: /* List of valuable user pages */ Added amazing user page by DrDisentangle

= List of valuable user pages =
I consider a user page "valuable" if it is up-to-date, and contains helpful information.

* [[User:Polygon/Better lower bound for BB(4,3)]]
* [[User:Polygon/Collection of BB Champions]]
* [[User:Polygon/Page for analyses]]
* [[User:XnoobSpeakable/Holdouts-temp]]
* [[User:Azerty/Busy Beaver Functions]]
* [[User:Autumn-Pan/Further-Reading]]
* [[User:Autumn-Pan/Busy-Beaver-Resource-Guide]]
* [[User:RobinCodes/Work on BB Domains]]
* [[User:RobinCodes/Next Steps]]
* [[User:RobinCodes/Machines at the Edge]]
* [[User:DrDisentangle/BB6 formal proofs]]

TMBR: April 2026

2026-04-12T06:11:55Z

RobinCodes: Better wording

TMBR: April 2026

2026-04-11T21:13:50Z

RobinCodes: Added TODO

TYBR: 2025

2026-04-11T21:12:54Z

RobinCodes: Added TODO to fix referencing

This is a summary research from all of 2025 (as documented in individual [[:Category:This Month in Beaver Research|This Month in Beaver Research]] throughout the year). 2025 was a very productive year for [[Busy Beaver Challenge|BBChallenge]]: about 60% of the next domain, [[BB(6)]], was solved. Furthermore, new champions were discovered for BB(6), [[BB(7)]] and [[BB(4,3)]]. Many models of computation other than Turing Machines were also explored - most notably [[Fractran]] and [[Instruction-Limited Busy Beaver]]. Some new methods were developed, such as [https://discord.com/channels/960643023006490684/1028746861395316776/1442964185599447152 mxdys's new version of FAR.]

This year, [[TMBR: November 2025#Themed Months|Themed Months]] were introduced - first, for [[BB(3,3)]], then for [[BB(2,5)]] - and the result is the clarification and verification of some of the results and techniques on the [https://discord.com/channels/960643023006490684/1259770474897080380 Discord] and [https://wiki.bbchallenge.org/wiki/BB(3,3) wiki]. See [[TMBR: November 2025#Themed Months]] for more information.

An annotated spreadsheet of [[BB(6)]] holdouts was also shared by [[User:RobinCodes|Robin Rovenszky]], which includes links to Discord discussions, classification of machines and is almost always up-to-date. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Google Sheets]

== This Year in Beaver Research (TYBR - "Thank You Beaver Researchers!") ==

=== Holdouts Reductions ===

{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|4319
|1326
|2993
|69.30%
|-
|[[BB(7)]]
|not enumerated
|20,387,509
|'''-'''
|'''-'''
|-
|[[BB(2,5)]]
|217
|75
|142
|65.44%
|-
|[[BB(2,6)]]
|22,302,296
|870,085
|21,432,211
|96.10%
|-
|[[BB(3,4)]]
|434,787,751
|12,435,284
|422,352,467
|97.14%
|-
|[[BB(4,3)]]
|460,916,384
|9,401,447
|451,514,937
|97.96%
|}

* [[BB(6)]] - Reduced from '''4319''' to '''1326''' [[holdouts]], a '''69.30% reduction.'''
* [[BB(2,5)]] - Reduced from '''217''' to '''75,''' a '''65.43% reduction.''' (The number of informal holdouts is '''64''').
* [[BB(7)]] - '''Enumeration was completed''', the number of holdouts was reduced from an initial 85,853,789 to '''20,387,509''' machines, a '''76.25%''' reduction.
* [[BB(4,3)]] - Reduced from 460,916,384 to 9,401,447 holdouts, a '''97.96% reduction.'''
* [[BB(3,4)]] - Reduced from 434,787,751 to 12,435,284 holdouts, a '''97.14% reduction.'''
* [[BB(2,6)]] - Reduced from 22,302,296 to '''870,085''' holdouts, a '''96.10%''' reduction.
* [[BB(2,7)]] - '''Enumeration started''', 100K of the 1M subtasks have been enumerated ('''10%''').

=== Champions ===

* [[BB(6)]] - On 16 June 2025, mxdys discovered {{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}, running for 10 ↑↑ 11010000 steps. This was surpassed on 25 June when mxdys discovered {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}, a TM which runs for <math>10 \uparrow\uparrow 10 \uparrow\uparrow 10 \uparrow\uparrow 8.10237</math> steps.
* [[BB(2,5)]] - [[1RB3LA4RB0RB2LA 1LB2LA3LA1RA1RZ|The champion]], initially discovered by Daniel Yuan on 24 Jun 2024 was [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 verified by mxdys] on 4 Jun 2025.
* [[BB(7)]] - Within three days of the start of the enumeration of BB(7), three champions were discovered. The first two were discovered by [[User:Sligocki|Shawn Ligocki]]: {{TM|1RB0RF_1LC0RE_1RD1LB_1LA1LD_0RA0LE_1RG0LB_1RZ1RB|halt}} with a sigma score of about 10 ↑↑ 22 and {{TM|1RB1RA_1RC0LC_0LD1LG_1LF0LE_1RZ1LF_0LA1LD_1RA1LC|halt}} with a sigma score of about 10 ↑↑ 35. This was followed by the discovery of {{TM|1RB0LG_1RC0RF_1LD1RZ_1LF0LE_1RA1LD_1LG1RE_0LB0LB|halt}}, achieving a sigma score of about 10 ↑↑ 46, by Terry Ligocki. On 10 May 2025, Pavel Kropitz discovered {{TM|1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF|halt}}, a TM which runs for over <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math> steps.
* [[BB(4,3)]] - In Feb 2025, Racheline identified {{TM|0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD|halt}} as the new BB(4,3) champion with a score of over <math>2 \uparrow\uparrow\uparrow 2^{2^{32}}</math>. In Oct 2025 [[User:Polygon|Polygon]] identified a new [[BB(4,3)]] and [[Maximum Consecutive Ones Function|num(4,3)]] champion with a score of over <math>10 \uparrow^{4} 4</math> ({{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}). These TMs were first proven to halt by Pavel Kropitz in May 2024, but their runtimes were not known at the time.
* [[Beeping Busy Beaver|BBB(3,3)]] - In March 2025 Nick Drozd [https://groups.google.com/g/busy-beaver-discuss/c/EuIXSir4Eps discovered] {{TM|1RB0LB2LA_1LA0RC0LB_2RC2RB0LC}}, which [[Quasihalt|quasihalts]] after running for more than 10 ↑↑ 6 steps.
* [[Busy Beaver for lambda calculus]] - The values of BBλ(31), BBλ(32), BBλ(33), BBλ(34), BBλ(35), BBλ(36) and BBλ(37) were solved, while new champions for BBλ(40), BBλ(41), BBλ(42), BBλ(43), BBλ(44), BBλ(45), BBλ(46), BBλ(48), BBλ(63) and BBλ(91) were discovered.
* [[Non-halting Turing machine#Translated cycler preperiod|BBS(4,3)]] - A new champion ({{TM|1RB1RD1LC_2LB1RB1LC_1LB1LA1LD_0RB2RA2RD}}) was discovered by changing the C0 transition of the [[BB(4,3)]] champion {{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}} from <code>C0 --> 1RZ</code> to <code>C0 --> 1LB</code>.
* [[Blanking Busy Beaver|BLB(3,3)]] - On 27 Dec 2025, [[User:Azerty|Azerty]] discovered {{TM|1RB1LA---_2RC2LB1RB_2LC2LA0RC}} which blanks the tape after 225 steps. A day later he found {{TM|1RB1RC---_1LB1RA2RB_0RB2LC0RC}}, which blanks the tape after 308 steps. That record was surpassed again on 31 Dec 2025 by the discovery of {{TM|1RB2LC2LA_1LC---2RA_2RC2LB0LC}} which blanks the tape after 329 steps.
* [[Non-halting Turing machine#Translated cycler period|BBP(3,3)]] - On 25 Dec 2025, [[User:Azerty|Azerty]] discovered {{TM|1RB2RC1LC_0RC0RB1LA_2LA2RC1LB}} which is a [[Translated cycler]] with a new record period length of 1195 steps.

=== New Methods ===

* New FAR using DFA generator by mxdys.[https://discord.com/channels/960643023006490684/1028746861395316776/1442964185599447152 <nowiki>[1]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1443990614483013632 <nowiki>[2]</nowiki>]
* @Bricks shared a method to estimate susceptibility to [[Block Analysis]] and a [https://docs.google.com/spreadsheets/d/1j00LBxxp9W7uz1wZdMIvDCZ56eReuH0IGO9Z8-yybcQ/edit?usp=sharing spreadsheet] of [[BB(6)]], [[BB(3,3)]] and [[BB(2,5)|BB(2,5]]) holdouts quantified by it.[https://discord.com/channels/960643023006490684/1239205785913790465/1430227817957953638 <nowiki>[3]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1430651610102632579 <nowiki>[4]</nowiki>]

TODO: Before July

=== Misc ===

* A fast algorithm for [[Consistent Collatz]] simulation was re-discovered and popularized. Using it,
** apgoucher simulated [[Antihydra]] to <math>2^{38}</math> iterations. This is actually a result from one year ago, but was rediscovered and added to the wiki. [https://discord.com/channels/960643023006490684/1026577255754903572/1271528180246773883 Source]
** [[User:Sligocki|Shawn Ligocki]] simulated {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC}} out to one additional Collatz reset, demonstrating that (if they halt, which they probviously should) they will have sigma scores <math>> 10^{10^{10^7}}</math>.
** This algorithm has near linear runtime (in the number of iterations simulated), but also linear memory growth since the parameters grow exponentially. This memory limit seems to be the main bottleneck to simulating Antihydra and other Consistent Collatz iterations further. There has been some discussion on more efficient memory usage or a distributed algorithm to support further scaling, but no results are available yet.
* Andrew Wade claims to have proven that BB(432) is [[Independence from ZFC|independent of ZF]]. [https://codeberg.org/ajwade/turing_machine_explorer Source]
* [[Piecewise Affine Function|Piecewise Affine Functions]] (PAF) were explored as a generalization of the [[BMO1]] rules:
** @Bard proved that 3 dimension PAF are [[Turing complete]].[https://discord.com/channels/960643023006490684/1239205785913790465/1420457986564030641]
** @star proved that 2 dimension PAF are Turing complete.[https://discord.com/channels/960643023006490684/1239205785913790465/1421271424588451915][https://discuss.bbchallenge.org/t/bmo1-type-problems-are-turing-complete/305]
** Shawn Ligocki wrote up a proof sketch that 2-region PAF are Turing complete.[https://discord.com/channels/960643023006490684/1239205785913790465/1422772752980639866]
** It was discovered that Amir Ben-Amram had already proven both of these results in 2015 (both the 2-dim and the 2-region results).
** BMO1 is a 2-dim, 2-region PAF so this provides some sense for the difficulty of the problem.
** This introduces a new type of [[Cryptids]] separate from previous [[Collatz-like]] ones.
* @coda [[TMBR: October 2025#Misc|shared a mechanical implementation]] of [[Antihydra]][https://discord.com/channels/960643023006490684/1362008236118511758/1425894649280598066] and @zts439 3d-printed a prototype.[https://discord.com/channels/960643023006490684/1362008236118511758/1427103960317296826]
* @vonhust created a fast TM simulator that averages 2 billion steps / s. It uses fixed-block [[Macro Machine|Macro Machines]] with each block bit-packed into integers. It is about 10x faster than direct simulators across most TMs.[https://discord.com/channels/960643023006490684/1226543091264126976/1438890558499061821]

TODO: Before July

=== BB Adjacent ===

* [[Instruction-Limited Busy Beaver]] was introduced and calculated up to BBi(7).
* [[Reversible Turing Machine]] Busy Beaver values were calculated up to BBrev(5).
* [[Terminating Turmite|Terminating Turmites]] (Relative Movement Turing Machines) were introduced and lower bounds for some small domains have been computed.
* John Tromp introduced the <math>BB \lambda _1(n)</math> function for [[Busy Beaver for lambda calculus#Oracle Busy Beaver|Busy Beaver for lambda calculus with an oracle]] and computed it up to <math>BB \lambda _1(22)</math>.
* Instruction-Limited Greedy Busy Beaver gBBi(n) and an [[Instruction-Limited Busy Beaver#Instruction-Limited Busy Beaver Variants|Instruction-Limited variant]] of the [[Blanking Busy Beaver]] (BLBi(n)) were introduced. gBBi(n) was computed up to n = 13 and BLBi(n) was computed up to n = 7.
* @savask shared the [[Bug Game]] (and fast-growing <math>Bug(H,W)</math> function).
* [[Fractran|Busy Beaver for Fractan]] (BBf) was introduced on 1 Nov by Jason Yuen.[https://discord.com/channels/960643023006490684/1362008236118511758/1433148101170040924] Exact values have been proven up to BBf(20) = 746[https://discord.com/channels/960643023006490684/1438019511155691521/1453213088630444168 <nowiki>[9]</nowiki>][https://discord.com/channels/960643023006490684/1438019511155691521/1452913055053778945 <nowiki>[10]</nowiki>] and exhaustive enumeration has been run up to size 22 (with BBf(21) > 31,957,632 and 397[https://discord.com/channels/960643023006490684/1438019511155691521/1453213088630444168 <nowiki>[11]</nowiki>] holdouts, and <math>BBf(22) > 1.146 \times 10^{62}</math> and 7807[https://discord.com/channels/960643023006490684/1438019511155691521/1453213088630444168 <nowiki>[12]</nowiki>] holdouts).
* [[Cyclic Tree Busy Beaver]] (CTBB) was introduced by @Jack on 14 Nov.[https://discord.com/channels/960643023006490684/1438694294042181742 <nowiki>[13]</nowiki>] The exact value is known for CTBB(2) = 5 and lower bounds have been found up to size 7 with CTBB(7) > 4↑↑↑↑(4↑↑↑3).
* [[Busy Beaver for lambda calculus#De Bruijn|Busy Beaver for lambda calculus using De Bruijn indexes]] was introduced with lower bounds having been calculated up to n = 34.
* [[Register machine|Busy Beaver for Register machines]] (MBB(n)) was introduced and computed up to n = 5 and a lower bound for n = 6 was discovered.

TODO: Before July and Semi-infinite tape Busy Beaver https://discord.com/channels/960643023006490684/1450894930422530161/1450894930422530161

=== In the News ===

* 6 January 2025. It Boltwise. [https://www.it-boltwise.de/durchbruch-im-busy-beaver-problem-eine-neue-aera-der-mathematik.html Durchbruch im Busy Beaver Problem: Eine neue Ära der Mathematik] (German) (English: Breakthrough in the Busy Beaver problem: A new era of mathematics).
* 24 March 2025. Nick Drozd. [https://nickdrozd.github.io/2025/03/24/bbb-3-3.html BBB(3, 3) > 10 ↑↑ 6].
* 21 Apr 2025. Shawn Ligocki. [https://www.sligocki.com/2025/04/21/lucys-moonlight.html Lucy's Moonlight: The 5% Champion].
* 9-13 June 2025. Terence Tao mentioned bbchallenge in their talk "The Equational Theories Project: advancing collaborative mathematical research at scale" ([https://www.youtube.com/watch?v=T4DE27uk0jw video] / [https://terrytao.wordpress.com/wp-content/uploads/2025/06/math-experiments.pdf slides]) at the [https://www.newton.ac.uk/event/bprw03/ 2025 Big Proof workshop]. The talk is about the [https://teorth.github.io/equational_theories/ Equational Theories Project], a large-scale mathematical collaboration that crowd-sourced a proof in Lean. Tao mentions bbchallenge as the only other example of a large-scale mathematical collaboration to prove a single result that he knows of.
* 28 June 2025. Scott Aaronson. [https://scottaaronson.blog/?p=8972 BusyBeaver(6) is really quite large].
* 1 July 2025. The Quanta Podcast. [https://discord.com/channels/960643023006490684/1285212639399776256/1389643208811745310 How Amateurs Solved a Major Computer Science Puzzle].
* 2 July 2025. Manon Bischoff. Spektrum. [https://www.spektrum.de/news/mathematik-die-sechste-fleissige-biber-zahl-ist-gigantisch/2274249 Wie der sechste Fleißige Biber die Mathematik an ihre Grenzen bringt].
* 3 July 2025. Nick Drozd. [https://nickdrozd.github.io/2025/07/03/busy-beaver-backwards.html Busy Beaver Backwards].
* 7 July 2025. Karmela Padavic-Callaghan. New Scientist. [https://www.newscientist.com/article/2487058-mathematicians-are-chasing-a-number-that-may-reveal-the-edge-of-maths/ Mathematicians are chasing a number that may reveal the edge of maths]. (Paywalled)
* 9 July 2025. David Roberts. [https://thehighergeometer.wordpress.com/2025/07/09/bb547176870-bb6-is-astronomically-larger/ BB(5)=47,176,870: BB(6) is … astronomically larger].
* 11 July 2025. New Scientist podcast [https://www.newscientist.com/podcasts/how-geoengineering-could-save-us-from-climate-disaster-have-we-broken-mathematics-why-exercise-reduces-cancer-risk/ episode 311]. Discusses mxdys's [[BB(6)]] pentation result "We’re brushing up against the edge of mathematics".
* 11 July 2025. Darren Orf. Popular Mechanics. [https://www.popularmechanics.com/science/math/a65357535/busy-beaver-six/ Mathematicians Say There’s a Number So Big, It’s Literally the Edge of Human Knowledge].
* 14 July 2025. Joe Brennan. Dario AS. [https://en.as.com/latest_news/meet-the-busy-beaver-number-a-number-so-huge-that-mathematicians-call-it-the-frontier-of-mathematical-knowledge-n/ Meet the Busy Beaver number, a number so huge that mathematicians call it the frontier of mathematical knowledge]
* 15 July 2025. Nick Drozd. [https://nickdrozd.github.io/2025/07/15/performance-hacks-for-bradys-algorithm.html Performance Hacks for Brady's Algorithm].
* 18 July 2025 https://francis.naukas.com/2025/07/18/espeluznante-nueva-cota-inferior-para-la-funcion-castor-afanoso-bb6/
* 22 Aug 2025. Ben Brubaker. Quanta Magazine. [https://www.quantamagazine.org/busy-beaver-hunters-reach-numbers-that-overwhelm-ordinary-math-20250822/ Busy Beaver Hunters Reach Numbers That Overwhelm Ordinary Math].
* 25-29 Aug 2025. [[User:Cosmo|Tristan Stérin]] presented [[:File:Conference poster for DNA31 by Tristan Stérin.png#file|a poster]] at [https://dna31.sciencesconf.org/ DNA 31].
* 1 Sep 2025. Katelyn Doucette. [https://katelyndoucette.com/articles/all-about-space-needle All About Space Needle].
* 12 Sep 2025. Katelyn Doucette. [https://katelyndoucette.com/articles/bugs-mazes-and-bradys-algorithm Bugs, Mazes, and the Unreasonably Effective Brady's Algorithm].
* 14 Sep 2025. Ben Brubaker. Wired. [https://www.wired.com/story/the-quest-to-find-the-longest-running-simple-computer-program/ The Quest to Find the Longest-Running Simple Computer Program]. (Reprint of Quanta article from last month).
* 17 Sep 2025. Hacker News. [https://news.ycombinator.com/item?id=45273999 Determination of the fifth Busy Beaver value].
* 18 Sep 2025. Tuomas Kangasniemi. Tekniikkatalous. [https://www.tekniikkatalous.fi/uutiset/a/85aafdaf-f506-4ce0-8035-a4dbe15ee4ff Iso matematiikan ongelma ratkesi 63 v jälkeen] (Finnish) (English: A big math problem solved after 63 years).
* 23 Sep 2025. Katelyn Doucette. [https://katelyndoucette.com/articles/building-the-busy-beaver-ladder Building the Busy Beaver Ladder].
* 30 Sep 2025. Nick Drozd. [https://nickdrozd.github.io/2025/09/30/shape-of-a-turing-machine.html The Shape of a Turing Machine].
* 22 Oct 2025. Ben Brubaker. [https://benbrubaker.com/why-busy-beaver-hunters-fear-the-antihydra/ Why Busy Beaver Hunters Fear the Antihydra]. ([https://news.ycombinator.com/item?id=45723359 Hacker News thread])
* 27 Oct 2025. [[User:Cosmo|Tristan Stérin]] gave a talk about [[bbchallenge]] and the [[BB(5)]] proof at Collège de France: [https://www.youtube.com/watch?v=YYrSdaB-6cE Le cinquième nombre Busy Beaver] (in French).[https://discord.com/channels/960643023006490684/1242208042460647575/1435724346051006516 <nowiki>[1]</nowiki>]
* 7-9 Nov 2025. Carl Kadie gave a talk on BB during the PyData Seattle 2025 conference: [https://www.youtube.com/watch?v=wSiF1Bm8f3s ''How to make Python programs run very slow (and new Turing Machine results)''].[https://discord.com/channels/960643023006490684/960643023530762343/1440090541936214017 <nowiki>[2]</nowiki>]
* 26 Dec 2025. New Scientist. [https://www.newscientist.com/article/2507465-mathematicians-spent-2025-exploring-the-edge-of-mathematics/ Mathematicians spent 2025 exploring the edge of mathematics]. (Paywalled)
* 31 Dec 2025. Nick Drozd. [https://nickdrozd.github.io/2025/12/31/goalposts.html Running out of places to move the goalposts to].

TODO: Fix referencing throughout the article
[[Category:This Year in Beaver Research|2025]]

TMBR: April 2026

2026-04-11T21:10:21Z

RobinCodes: /* BB Adjacent */ Added TODO

TMBR: April 2026

2026-04-11T21:09:42Z

RobinCodes: Updated references for consistency

TMBR: February 2026

2026-04-11T21:07:49Z

RobinCodes: Updated references for consistency and fixed grammatical error

{{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 Turing machines whose halting status is independent of the theories of [https://en.wikipedia.org/w/index.php?title=Peano_Arithmetic Peano Arithmetic] (372-state)[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 <nowiki>[1]</nowiki>] and ZFC paired with the axiom "There exist arbitrarily large [[wikipedia:Subtle_cardinal|subtle cardinals]]" (493-state)[https://discord.com/channels/960643023006490684/1474221209494491198/1476249816488087572 <nowiki>[2]</nowiki>]. 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 Turing machines, whose halting status is independent of the theories of [https://en.wikipedia.org/w/index.php?title=Peano_Arithmetic Peano Arithmetic] (BB(372)[https://discord.com/channels/960643023006490684/1466652214247559198/1471186212743155856 <nowiki>[3]</nowiki>] and ZFC+"There exist arbitrarily large [[wikipedia:Subtle_cardinal|subtle cardinals]]" (BB(493))[https://discord.com/channels/960643023006490684/1474221209494491198/1476249816488087572 <nowiki>[4]</nowiki>] (see [[Logical independence]] for more context and the machines)
* 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>[3 machines]</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>[5]</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>[6]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1475911180965904577 <nowiki>[7]</nowiki>][https://discord.com/channels/960643023006490684/1239205785913790465/1477040884728987820 <nowiki>[8]</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 to '''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>[9]</nowiki>]

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

TMBR: January 2026

2026-04-11T21:01:48Z

RobinCodes: Updated references for consistency

For a summary of our research in 2025, see [[TYBR: 2025]].{{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(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 <nowiki>[3]</nowiki>]

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