BusyBeaverWiki - User contributions [en]

TMBR: April 2026

2026-04-09T07:56:11Z

Qwertyasdf: 2x6 reduction

{{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. TODO: Write proper introductory paragraph. Important: BB(2,5) formalisation result.

== Misc ==
* Katelyn Doucette completed [https://github.com/Laturas/FractranVisualizer a visualizer] for Fractran space-time diagrams.[https://discord.com/channels/960643023006490684/1438019511155691521/1488727841951449088 <nowiki>[1]</nowiki>]

== Holdouts ==

* [[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 holdout count to 67, 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.
*[[BB(2,6)]]
**Andrew Ducharme [https://discord.com/channels/960643023006490684/1084047886494470185/1491652128123388026 reduced] the number of holdouts from 545,005 to '''542,325''' via Enumerate.py, a 0.49% reduction.
*[[BB(6)]]
**Discord user sheep discovered {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}, a possible [[cryptid]] similar to [[Space Needle]].

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

TMBR: April 2026

2026-04-07T11:57:48Z

Qwertyasdf: add new possible cryptid discovery

{{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. TODO: Write proper introductory paragraph. Important: BB(2,5) formalisation result.

== Misc ==
* Katelyn Doucette completed [https://github.com/Laturas/FractranVisualizer a visualizer] for Fractran space-time diagrams.[https://discord.com/channels/960643023006490684/1438019511155691521/1488727841951449088 <nowiki>[1]</nowiki>]

== Holdouts ==

* [[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 holdout count to 67, 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.
*[[BB(6)]]
**Discord user sheep discovered a possible cryptid {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}} similar to [[Space Needle]].

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

TMBR: March 2026

2026-03-27T01:19:38Z

Qwertyasdf: /* Champions */ add new holdout list

{{TMBRnav|February 2026|April 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 March 2026. We celebrated bbchallenge's fourth birthday on 8 March.

TODO: Write a proper introductory paragraph.

== Champions ==

* Discord user 50_ft_lock [https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 found] a new BB(13) [[Champions|champion]] which surpasses [[Graham's number]], reducing the upper bound of Graham-beating TMs to 13 states.

== Meta ==

* Famous math Youtuber [https://www.youtube.com/@twoswap 2swap] made a [https://discord.com/channels/960643023006490684/1362008236118511758/1478973587653136456 couple of videos about Turing Machines] arranged into grids and colored based on their halting status for BB(2,2), BB(3,2), BB(2,3), BB(4,2) and BB(5,2) respectively, then made a [https://www.youtube.com/watch?v=1BI3qItCJ2M two-hour long Youtube video] about the same topic on [https://www.youtube.com/@3cycle their second channel].

== BB Adjacent ==
* [[Fractran]]: In BBf(21), Claude Opus 4.6 gave a proof that all 140 holdouts do not halt. This tentatively proves that BBf(21) = 31,957,632.
* [[Fractran]]: A Cryptid was discovered in BBf(22) with the help of Claude Opus 4.6.

== Holdouts ==

* [[BB(6)]]: 1e14 machines: '''171'''. 1e15 machines: '''237'''. '''12 solved machines'''. TODO: Update
** Holdouts lists for machines not simulated to '''1e14 and 1e15''' steps were created. The holdouts list counts were '''178 and 252''' respectively. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet for BB6].
** Later, [https://discord.com/channels/960643023006490684/1477591686514212894/1478401591307407554 prurq found 10 more machines] in the holdouts list that had previously been simulated to 1e15: thus the new 1e15 holdout count was '''242.'''
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1478626634729914450 solved two machines] using FAR.
** prurq found two machines [https://discord.com/channels/960643023006490684/1239205785913790465/1478518151393312822][https://discord.com/channels/960643023006490684/1239205785913790465/1478525451826888775] to be [[Translated Cycler|Translated Cyclers]]. Shawn Ligocki [https://discord.com/channels/960643023006490684/1239205785913790465/1478781268664778989 verified one of them] and discovered its preperiod to be over <math>10^{12}</math> steps. mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1478827653636296850 verified the other]. Both were 1e14 and 1e15 holdouts, thus reducing those holdout counts by 2.
** mxdys found four more [https://discord.com/channels/960643023006490684/1239205785913790465/1478734570672095333][https://discord.com/channels/960643023006490684/1239205785913790465/1479045001089384600] Translated Cyclers in the remaining holdouts and solved [https://discord.com/channels/960643023006490684/1239205785913790465/1478828100707025151 one more TM] using FAR.
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1479313219003748484 solved a machine] using FAR. This machine was a 1e14 and 1e15 holdout, thus reducing those holdout counts by 1.
** [https://discord.com/channels/960643023006490684/1477591686514212894/1480544451033170060 prurq found 3 more machines previously simulated so far] in the 1e14 list, and 2 more in the 1e15 list (both were also 1e14). This means 1e14 holdout count was reduced by 3, and 1e15 holdout count was reduced by 2.
** Discord user mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1480686400067342346 simulated a 1e14 holdout thus far], therefore reducing that holdout count by 1.
** prurq [https://discord.com/channels/960643023006490684/1471178503235043493/1481022684975337586 found a machine to be a Translated Cycler].
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1481557945346035814 decided a machine] using FAR.
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1486207538843222116 released] a new holdout list of 1161 machines.
*[[BB(7)]]:
**Andrew Ducharme [https://discord.com/channels/960643023006490684/1369339127652159509/1481889113232904295 reduced] the number of holdouts from 18,195,192 to 18,036,852 (a 0.87% reduction) via the mxdys FAR decider.
*[[BB(2,5)]]:
**[https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 Peacemaker II solved a machine using FAR] and mxdys confirmed two[https://discord.com/channels/960643023006490684/1259770421046411285/1483043448855461989][https://discord.com/channels/960643023006490684/1259770421046411285/1483043657778069564] machines to be non-halting. Thus, the new holdout count is 69, or 60 considering informal proofs.
*[[BB(3,3)]]:
**mxdys [https://discord.com/channels/960643023006490684/1259770474897080380/1482680295357677651 formalised] three remaining informal results (650, 412, 279) into Rocq.

[[Category:This Month in Beaver Research|2026-03]]

TMBR: March 2026

2026-03-15T15:28:16Z

Qwertyasdf: grammar

{{TMBRnav|February 2026|April 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 March 2026. We celebrated bbchallenge's fourth birthday on 8 March.

TODO: Write a proper introductory paragraph.

== Champions ==

* Discord user 50_ft_lock [https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 found] a new BB(13) [[Champions|champion]] which surpasses [[Graham's number]], reducing the upper bound of Graham-beating TMs to 13 states.

== Meta ==

* Famous math youtuber [https://www.youtube.com/@twoswap 2swap] made a [https://discord.com/channels/960643023006490684/1362008236118511758/1478973587653136456 couple of videos about Turing Machines] arranged into grids and colored based on their halting status for BB(2,2), BB(3,2), BB(2,3), BB(4,2) and BB(5,2) respectively, then made a [https://www.youtube.com/watch?v=1BI3qItCJ2M two-hour long Youtube video] about the same topic on [https://www.youtube.com/@3cycle their second channel].

== Holdouts ==

* [[BB(6)]]: 1e14 machines: '''171'''. 1e15 machines: '''237'''. '''12 solved machines'''.
** Holdouts lists for machines not simulated to '''1e14 and 1e15''' steps were created. The holdouts list counts were '''178 and 252''' respectively. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet for BB6].
** Later, [https://discord.com/channels/960643023006490684/1477591686514212894/1478401591307407554 prurq found 10 more machines] in the holdouts list that had previously been simulated to 1e15: thus the new 1e15 holdout count was '''242.'''
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1478626634729914450 solved two machines] using FAR.
** prurq found two machines [https://discord.com/channels/960643023006490684/1239205785913790465/1478518151393312822][https://discord.com/channels/960643023006490684/1239205785913790465/1478525451826888775] to be [[Translated Cycler|Translated Cyclers]]. Shawn Ligocki [https://discord.com/channels/960643023006490684/1239205785913790465/1478781268664778989 verified one of them] and discovered its preperiod to be over <math>10^{12}</math> steps. mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1478827653636296850 verified the other]. Both were 1e14 and 1e15 holdouts, thus reducing those holdout counts by 2.
** mxdys found four more [https://discord.com/channels/960643023006490684/1239205785913790465/1478734570672095333][https://discord.com/channels/960643023006490684/1239205785913790465/1479045001089384600] Translated Cyclers in the remaining holdouts and solved [https://discord.com/channels/960643023006490684/1239205785913790465/1478828100707025151 one more TM] using FAR.
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1479313219003748484 solved a machine] using FAR. This machine was a 1e14 and 1e15 holdout, thus reducing those holdout counts by 1.
** [https://discord.com/channels/960643023006490684/1477591686514212894/1480544451033170060 prurq found 3 more machines previously simulated so far] in the 1e14 list, and 2 more in the 1e15 list (both were also 1e14). This means 1e14 holdout count was reduced by 3, and 1e15 holdout count was reduced by 2.
** Discord user mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1480686400067342346 simulated a 1e14 holdout thus far], therefore reducing that holdout count by 1.
** prurq [https://discord.com/channels/960643023006490684/1471178503235043493/1481022684975337586 found a machine to be a Translated Cycler].
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1481557945346035814 decided a machine] using FAR.
*[[BB(7)]]:
**Andrew Ducharme [https://discord.com/channels/960643023006490684/1369339127652159509/1481889113232904295 reduced] the number of holdouts from 18,195,192 to 18,036,852 (a 0.87% reduction) via the mxdys FAR decider.
*[[BB(2,5)]]:
**[https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 Peacemaker II solved a machine using FAR.] Thus, the new holdout count is 71, or 60 considering informal proofs.
*[[BB(3,3)]]:
**mxdys [https://discord.com/channels/960643023006490684/1259770474897080380/1482680295357677651 formalised] three remaining informal results (650, 412, 279) into Rocq.

[[Category:This Month in Beaver Research|2026-03]]

BB(3,3)

2026-03-15T15:25:05Z

Qwertyasdf: remove now-outdated sentence

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

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

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

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

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

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

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

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

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

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

On [https://discord.com/channels/960643023006490684/1259770474897080380/1344554618188730368 26 Feb 2025], @mxdys published a list of 19 holdouts that withstood state-of-the-art Rocq deciders. Some of these machines already had fairly rigorous or even full Rocq proofs for non-halting, which were integrated into a 12 TM Rocq holdout list published on [https://discord.com/channels/960643023006490684/1259770474897080380/1409402854292066335 24 Aug 2025.] Then, on 27 Aug 2025, @mxdys certified 2 more machines in Rocq, lowering the Rocq holdout list to 10 TMs. Afterwards, on [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 30 Aug 2025], @mxdys certified one more machine in Rocq, lowering the Rocq holdout list to 9 TMs. All 3 of the machines that have a moderate rigor arguments are believed to never halt due to [[Longitudinal Analysis|longitudinal analysis]] by @Legion.

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

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

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

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

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

On 16 March 2026, @mxdys [https://discord.com/channels/960643023006490684/1259770474897080380/1482680295357677651 formalised] into Rocq the following informal results:

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

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

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

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

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

BB(3,3)

2026-03-15T15:17:27Z

Qwertyasdf: /* Holdouts */ descriptor

The 3-state, 3-symbol Busy Beaver problem, '''BB(3,3)''', is unsolved. With the discovery of the [[Cryptids|Cryptid]] machine [[Bigfoot]] in October 2023, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(3,3) and thus [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html BB(3,3) is Hard]. There are nine holdouts unproven in Rocq, three of which are believed to never halt. Of the remaining six with unknown behavior, there are two pairs of holdouts with equivalent behavior, so, in effect, there are four holdouts remaining.

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

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

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

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

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

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

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

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

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

On [https://discord.com/channels/960643023006490684/1259770474897080380/1344554618188730368 26 Feb 2025], @mxdys published a list of 19 holdouts that withstood state-of-the-art Rocq deciders. Some of these machines already had fairly rigorous or even full Rocq proofs for non-halting, which were integrated into a 12 TM Rocq holdout list published on [https://discord.com/channels/960643023006490684/1259770474897080380/1409402854292066335 24 Aug 2025.] Then, on 27 Aug 2025, @mxdys certified 2 more machines in Rocq, lowering the Rocq holdout list to 10 TMs. Afterwards, on [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 30 Aug 2025], @mxdys certified one more machine in Rocq, lowering the Rocq holdout list to 9 TMs. All 3 of the machines that have a moderate rigor arguments are believed to never halt due to [[Longitudinal Analysis|longitudinal analysis]] by @Legion.

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

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

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

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

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

On 16 March 2026, @mxdys [https://discord.com/channels/960643023006490684/1259770474897080380/1482680295357677651 formalised] into Rocq the following informal results:

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

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

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

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

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

TMBR: March 2026

2026-03-15T15:16:30Z

Qwertyasdf: /* Holdouts */ clarify

{{TMBRnav|February 2026|April 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 March 2026. We celebrated bbchallenge's fourth birthday on 8 March.

TODO: Write a proper introductory paragraph.

== Champions ==

* Discord user 50_ft_lock [https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 found] a new BB(13) [[Champions|champion]], which surpasses [[Graham's number]], reducing the upper bound of Graham-beating TMs to 13 states.

== Meta ==

* Famous math youtuber [https://www.youtube.com/@twoswap 2swap] made a [https://discord.com/channels/960643023006490684/1362008236118511758/1478973587653136456 couple of videos about Turing Machines] arranged into grids and colored based on their halting status for BB(2,2), BB(3,2), BB(2,3), BB(4,2) and BB(5,2) respectively, then made a [https://www.youtube.com/watch?v=1BI3qItCJ2M two-hour long Youtube video] about the same topic on [https://www.youtube.com/@3cycle their second channel].

== Holdouts ==

* [[BB(6)]]: 1e14 machines: '''171'''. 1e15 machines: '''237'''. '''12 solved machines'''.
** Holdouts lists for machines not simulated to '''1e14 and 1e15''' steps were created. The holdouts list counts were '''178 and 252''' respectively. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet for BB6].
** Later, [https://discord.com/channels/960643023006490684/1477591686514212894/1478401591307407554 prurq found 10 more machines] in the holdouts list that had previously been simulated to 1e15: thus the new 1e15 holdout count was '''242.'''
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1478626634729914450 solved two machines] using FAR.
** prurq found two machines [https://discord.com/channels/960643023006490684/1239205785913790465/1478518151393312822][https://discord.com/channels/960643023006490684/1239205785913790465/1478525451826888775] to be [[Translated Cycler|Translated Cyclers]]. Shawn Ligocki [https://discord.com/channels/960643023006490684/1239205785913790465/1478781268664778989 verified one of them] and discovered its preperiod to be over <math>10^{12}</math> steps. mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1478827653636296850 verified the other]. Both were 1e14 and 1e15 holdouts, thus reducing those holdout counts by 2.
** mxdys found four more [https://discord.com/channels/960643023006490684/1239205785913790465/1478734570672095333][https://discord.com/channels/960643023006490684/1239205785913790465/1479045001089384600] Translated Cyclers in the remaining holdouts and solved [https://discord.com/channels/960643023006490684/1239205785913790465/1478828100707025151 one more TM] using FAR.
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1479313219003748484 solved a machine] using FAR. This machine was a 1e14 and 1e15 holdout, thus reducing those holdout counts by 1.
** [https://discord.com/channels/960643023006490684/1477591686514212894/1480544451033170060 prurq found 3 more machines previously simulated so far] in the 1e14 list, and 2 more in the 1e15 list (both were also 1e14). This means 1e14 holdout count was reduced by 3, and 1e15 holdout count was reduced by 2.
** Discord user mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1480686400067342346 simulated a 1e14 holdout thus far], therefore reducing that holdout count by 1.
** prurq [https://discord.com/channels/960643023006490684/1471178503235043493/1481022684975337586 found a machine to be a Translated Cycler].
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1481557945346035814 decided a machine] using FAR.
*[[BB(7)]]:
**Andrew Ducharme [https://discord.com/channels/960643023006490684/1369339127652159509/1481889113232904295 reduced] the number of holdouts from 18,195,192 to 18,036,852 (a 0.87% reduction) via the mxdys FAR decider.
*[[BB(2,5)]]:
**[https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 Peacemaker II solved a machine using FAR.] Thus, the new holdout count is 71, or 60 considering informal proofs.
*[[BB(3,3)]]:
**mxdys [https://discord.com/channels/960643023006490684/1259770474897080380/1482680295357677651 formalised] three remaining informal results (650, 412, 279) into Rocq.

[[Category:This Month in Beaver Research|2026-03]]

BB(3,3)

2026-03-15T15:16:24Z

Qwertyasdf: /* Solved with moderate rigor */ upt mxdys formalisation

The 3-state, 3-symbol Busy Beaver problem, '''BB(3,3)''', is unsolved. With the discovery of the [[Cryptids|Cryptid]] machine [[Bigfoot]] in October 2023, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(3,3) and thus [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html BB(3,3) is Hard]. There are nine holdouts unproven in Rocq, three of which are believed to never halt. Of the remaining six with unknown behavior, there are two pairs of holdouts with equivalent behavior, so, in effect, there are four holdouts remaining.

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

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

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

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

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

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

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

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

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

On [https://discord.com/channels/960643023006490684/1259770474897080380/1344554618188730368 26 Feb 2025], @mxdys published a list of 19 holdouts that withstood state-of-the-art Rocq deciders. Some of these machines already had fairly rigorous or even full Rocq proofs for non-halting, which were integrated into a 12 TM Rocq holdout list published on [https://discord.com/channels/960643023006490684/1259770474897080380/1409402854292066335 24 Aug 2025.] Then, on 27 Aug 2025, @mxdys certified 2 more machines in Rocq, lowering the Rocq holdout list to 10 TMs. Afterwards, on [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 30 Aug 2025], @mxdys certified one more machine in Rocq, lowering the Rocq holdout list to 9 TMs. All 3 of the machines that have a moderate rigor arguments are believed to never halt due to [[Longitudinal Analysis|longitudinal analysis]] by @Legion.

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

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

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

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

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

On 16 March 2026, @mxdys [https://discord.com/channels/960643023006490684/1259770474897080380/1482680295357677651 formalised] into Rocq the following informal results:

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

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

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

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

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

TMBR: March 2026

2026-03-15T15:05:41Z

Qwertyasdf: /* Holdouts */ add 3,3 formalisations

{{TMBRnav|February 2026|April 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 March 2026. We celebrated bbchallenge's fourth birthday on 8 March.

TODO: Write a proper introductory paragraph.

== Champions ==

* Discord user 50_ft_lock [https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 found] a new BB(13) [[Champions|champion]], which surpasses [[Graham's number]], reducing the upper bound of Graham-beating TMs to 13 states.

== Meta ==

* Famous math youtuber [https://www.youtube.com/@twoswap 2swap] made a [https://discord.com/channels/960643023006490684/1362008236118511758/1478973587653136456 couple of videos about Turing Machines] arranged into grids and colored based on their halting status for BB(2,2), BB(3,2), BB(2,3), BB(4,2) and BB(5,2) respectively, then made a [https://www.youtube.com/watch?v=1BI3qItCJ2M two-hour long Youtube video] about the same topic on [https://www.youtube.com/@3cycle their second channel].

== Holdouts ==

* [[BB(6)]]: 1e14 machines: '''171'''. 1e15 machines: '''237'''. '''12 solved machines'''.
** Holdouts lists for machines not simulated to '''1e14 and 1e15''' steps were created. The holdouts list counts were '''178 and 252''' respectively. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet for BB6].
** Later, [https://discord.com/channels/960643023006490684/1477591686514212894/1478401591307407554 prurq found 10 more machines] in the holdouts list that had previously been simulated to 1e15: thus the new 1e15 holdout count was '''242.'''
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1478626634729914450 solved two machines] using FAR.
** prurq found two machines [https://discord.com/channels/960643023006490684/1239205785913790465/1478518151393312822][https://discord.com/channels/960643023006490684/1239205785913790465/1478525451826888775] to be [[Translated Cycler|Translated Cyclers]]. Shawn Ligocki [https://discord.com/channels/960643023006490684/1239205785913790465/1478781268664778989 verified one of them] and discovered its preperiod to be over <math>10^{12}</math> steps. mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1478827653636296850 verified the other]. Both were 1e14 and 1e15 holdouts, thus reducing those holdout counts by 2.
** mxdys found four more [https://discord.com/channels/960643023006490684/1239205785913790465/1478734570672095333][https://discord.com/channels/960643023006490684/1239205785913790465/1479045001089384600] Translated Cyclers in the remaining holdouts and solved [https://discord.com/channels/960643023006490684/1239205785913790465/1478828100707025151 one more TM] using FAR.
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1479313219003748484 solved a machine] using FAR. This machine was a 1e14 and 1e15 holdout, thus reducing those holdout counts by 1.
** [https://discord.com/channels/960643023006490684/1477591686514212894/1480544451033170060 prurq found 3 more machines previously simulated so far] in the 1e14 list, and 2 more in the 1e15 list (both were also 1e14). This means 1e14 holdout count was reduced by 3, and 1e15 holdout count was reduced by 2.
** Discord user mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1480686400067342346 simulated a 1e14 holdout thus far], therefore reducing that holdout count by 1.
** prurq [https://discord.com/channels/960643023006490684/1471178503235043493/1481022684975337586 found a machine to be a Translated Cycler].
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1481557945346035814 decided a machine] using FAR.
*[[BB(7)]]:
**Andrew Ducharme [https://discord.com/channels/960643023006490684/1369339127652159509/1481889113232904295 reduced] the number of holdouts from 18,195,192 to 18,036,852 (a 0.87% reduction) via the mxdys FAR decider.
*[[BB(2,5)]]:
**[https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 Peacemaker II solved a machine using FAR.] Thus, the new holdout count is 71, or 60 considering informal proofs.
*[[BB(3,3)]]:
**mxdys [https://discord.com/channels/960643023006490684/1259770474897080380/1482680295357677651 formalised] three informal results (650, 412, 279) into Rocq.

[[Category:This Month in Beaver Research|2026-03]]

TMBR: March 2026

2026-03-13T07:43:28Z

Qwertyasdf: /* Holdouts */ add bb7 reduction

{{TMBRnav|February 2026|April 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 March 2026. We celebrated bbchallenge's fourth birthday on 8 March.

TODO: Write a proper introductory paragraph.

== Champions ==

* Discord user 50_ft_lock [https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 found] a new BB(13) [[Champions|champion]], which surpasses [[Graham's number]], reducing the upper bound of Graham-beating TMs to 13 states.

== Meta ==

* Famous math youtuber [https://www.youtube.com/@twoswap 2swap] made a [https://discord.com/channels/960643023006490684/1362008236118511758/1478973587653136456 couple of videos about Turing Machines] arranged into grids and colored based on their halting status for BB(2,2), BB(3,2), BB(2,3), BB(4,2) and BB(5,2) respectively, then made a [https://www.youtube.com/watch?v=1BI3qItCJ2M two-hour long Youtube video] about the same topic on [https://www.youtube.com/@3cycle their second channel].

== Holdouts ==

* [[BB(6)]]: 1e14 machines: '''171'''. 1e15 machines: '''237'''. '''12 solved machines'''.
** Holdouts lists for machines not simulated to '''1e14 and 1e15''' steps were created. The holdouts list counts were '''178 and 252''' respectively. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet for BB6].
** Later, [https://discord.com/channels/960643023006490684/1477591686514212894/1478401591307407554 prurq found 10 more machines] in the holdouts list that had previously been simulated to 1e15: thus the new 1e15 holdout count was '''242.'''
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1478626634729914450 solved two machines] using FAR.
** prurq found two machines [https://discord.com/channels/960643023006490684/1239205785913790465/1478518151393312822][https://discord.com/channels/960643023006490684/1239205785913790465/1478525451826888775] to be [[Translated Cycler|Translated Cyclers]]. Shawn Ligocki [https://discord.com/channels/960643023006490684/1239205785913790465/1478781268664778989 verified one of them] and discovered its preperiod to be over <math>10^{12}</math> steps. mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1478827653636296850 verified the other]. Both were 1e14 and 1e15 holdouts, thus reducing those holdout counts by 2.
** mxdys found four more [https://discord.com/channels/960643023006490684/1239205785913790465/1478734570672095333][https://discord.com/channels/960643023006490684/1239205785913790465/1479045001089384600] Translated Cyclers in the remaining holdouts and solved [https://discord.com/channels/960643023006490684/1239205785913790465/1478828100707025151 one more TM] using FAR.
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1479313219003748484 solved a machine] using FAR. This machine was a 1e14 and 1e15 holdout, thus reducing those holdout counts by 1.
** [https://discord.com/channels/960643023006490684/1477591686514212894/1480544451033170060 prurq found 3 more machines previously simulated so far] in the 1e14 list, and 2 more in the 1e15 list (both were also 1e14). This means 1e14 holdout count was reduced by 3, and 1e15 holdout count was reduced by 2.
** Discord user mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1480686400067342346 simulated a 1e14 holdout thus far], therefore reducing that holdout count by 1.
** prurq [https://discord.com/channels/960643023006490684/1471178503235043493/1481022684975337586 found a machine to be a Translated Cycler].
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1481557945346035814 decided a machine] using FAR.
*[[BB(7)]]:
**Andrew Ducharme [https://discord.com/channels/960643023006490684/1369339127652159509/1481889113232904295 reduced] the number of holdouts from 18,195,192 to 18,036,852 (a 0.87% reduction) via the mxdys FAR decider.
*[[BB(2,5)]]:
**[https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 Peacemaker II solved a machine using FAR.] Thus, the new holdout count is 71, or 60 considering informal proofs.

[[Category:This Month in Beaver Research|2026-03]]

TMBR: March 2026

2026-03-13T07:36:53Z

Qwertyasdf: add new bb13 champion

{{TMBRnav|February 2026|April 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 March 2026. We celebrated bbchallenge's fourth birthday on 8 March.

TODO: Write a proper introductory paragraph.

== Champions ==

* Discord user 50_ft_lock [https://discord.com/channels/960643023006490684/1331570843829932063/1481871400640839691 found] a new BB(13) [[Champions|champion]], which surpasses [[Graham's number]], reducing the upper bound of Graham-beating TMs to 13 states.

== Meta ==

* Famous math youtuber [https://www.youtube.com/@twoswap 2swap] made a [https://discord.com/channels/960643023006490684/1362008236118511758/1478973587653136456 couple of videos about Turing Machines] arranged into grids and colored based on their halting status for BB(2,2), BB(3,2), BB(2,3), BB(4,2) and BB(5,2) respectively, then made a [https://www.youtube.com/watch?v=1BI3qItCJ2M two-hour long Youtube video] about the same topic on [https://www.youtube.com/@3cycle their second channel].

== Holdouts ==

* [[BB(6)]]: 1e14 machines: '''171'''. 1e15 machines: '''237'''. '''12 solved machines'''.
** Holdouts lists for machines not simulated to '''1e14 and 1e15''' steps were created. The holdouts list counts were '''178 and 252''' respectively. See [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 Spreadsheet for BB6].
** Later, [https://discord.com/channels/960643023006490684/1477591686514212894/1478401591307407554 prurq found 10 more machines] in the holdouts list that had previously been simulated to 1e15: thus the new 1e15 holdout count was '''242.'''
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1478626634729914450 solved two machines] using FAR.
** prurq found two machines [https://discord.com/channels/960643023006490684/1239205785913790465/1478518151393312822][https://discord.com/channels/960643023006490684/1239205785913790465/1478525451826888775] to be [[Translated Cycler|Translated Cyclers]]. Shawn Ligocki [https://discord.com/channels/960643023006490684/1239205785913790465/1478781268664778989 verified one of them] and discovered its preperiod to be over <math>10^{12}</math> steps. mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1478827653636296850 verified the other]. Both were 1e14 and 1e15 holdouts, thus reducing those holdout counts by 2.
** mxdys found four more [https://discord.com/channels/960643023006490684/1239205785913790465/1478734570672095333][https://discord.com/channels/960643023006490684/1239205785913790465/1479045001089384600] Translated Cyclers in the remaining holdouts and solved [https://discord.com/channels/960643023006490684/1239205785913790465/1478828100707025151 one more TM] using FAR.
** Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1479313219003748484 solved a machine] using FAR. This machine was a 1e14 and 1e15 holdout, thus reducing those holdout counts by 1.
** [https://discord.com/channels/960643023006490684/1477591686514212894/1480544451033170060 prurq found 3 more machines previously simulated so far] in the 1e14 list, and 2 more in the 1e15 list (both were also 1e14). This means 1e14 holdout count was reduced by 3, and 1e15 holdout count was reduced by 2.
** Discord user mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1480686400067342346 simulated a 1e14 holdout thus far], therefore reducing that holdout count by 1.
** prurq [https://discord.com/channels/960643023006490684/1471178503235043493/1481022684975337586 found a machine to be a Translated Cycler].
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1481557945346035814 decided a machine] using FAR.
*[[BB(2,5)]]:
**[https://discord.com/channels/960643023006490684/1259770421046411285/1481197573611061311 Peacemaker II solved a machine using FAR.] Thus, the new holdout count is 71, or 60 considering informal proofs.

[[Category:This Month in Beaver Research|2026-03]]

TMBR: February 2026

2026-02-27T22:30:43Z

Qwertyasdf: /* Holdouts */ upd

{{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.

== 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 ==
TODO: independence from Peano (Legion) (see [[Logical independence]])

TODO: prurq new fast simulation method (see [https://discord.com/channels/960643023006490684/1471178503235043493 Discord thread])

TODO: "Cascade" (see [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 Discord thread] )

== 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 ==
*[[BB(2,5)]]: '''2 solved machines.'''
**Andrew Ducharme found a machine nonhalting on [https://discord.com/channels/960643023006490684/1259770421046411285/1471227102844944510 11 Feb] via the newly released mxdys FAR decider. This was verified in Rocq by mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471228798505582602 the same day].
**mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471229409829847111 announced another TM proven the same day], which turned out to be a translated cycler.
**Peacemaker II [https://discord.com/channels/960643023006490684/1259770421046411285/1472647706835943596 found the high-level behaviour of a machine], which turned out to be a relatively simple-to-describe string rewriting problem of sorts.
*[[BB(6)]]: '''XX''' machines simulated to 1e13, '''XX''' solved machines. TODO: Update
**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 newly released mxdys FAR decider.
**Alistaire [https://discord.com/channels/960643023006490684/1239205785913790465/1472376779825090713 found a machine nonhalting] using Quick_Sim.py.
**prurq simulated 38 machines for >1e13 steps[https://discord.com/channels/960643023006490684/1471178503235043493/1471486886890704967 <nowiki>[19 machines]</nowiki>][https://docs.google.com/spreadsheets/d/1zMhtW_edMxrfUry-hVMFsDg3T1p_udjC2V2RKu6oSKE/edit?usp=sharing <nowiki>[19 more machines]</nowiki>] with his new method [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 "Cascade".]
**Alistaire [https://discord.com/channels/960643023006490684/1239205785913790465/1472246267345113158 simulated 13 machines] for >1e13 steps, 6 of which had already been simulated by prurq, essentialy double-verifying them.
**Discord user @mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1472325414344069271 simulated a TM] for >1e13 steps, which also turned out to have been simulated by prurq already.
**Afterall, the informal holdout count is '''1299''', and the formal holdout count is 1303 (8 newly solved machines were solved by highly trusted code, 4 informal TMs beforehand). Rocq-verified holdout count is 1314. There remain '''160''' machines to be simulated up to 1e13. TODO: Update
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1473950417275850804 released] a [[holdouts list]] of 1226 machines up to equivalence, some of which were decided via [https://discord.com/channels/960643023006490684/1226543091264126976/1469937272752177298 new mxdys method for longitudinal acceleration].
**Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1474302212284092436 found 9 non-halting machines] in that list using the mxdys FAR decider.[https://discord.com/channels/960643023006490684/1239205785913790465/1474302212284092436][https://discord.com/channels/960643023006490684/1239205785913790465/1475911180965904577][https://discord.com/channels/960643023006490684/1239205785913790465/1477040884728987820]
*[[BB(7)]]:
**Andrew Ducharme has reduced the number of holdouts from 19,303,801 to 18,254,545 (a 5.44% reduction) and then '''18,195,192''' (0.33%) using the newly released mxdys 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 FAR decider.
***Another 0.47% reduction left 548,993 holdouts.[https://discord.com/channels/960643023006490684/1084047886494470185/1475216024734269644]

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

TMBR: February 2026

2026-02-25T02:24:36Z

Qwertyasdf: /* Holdouts */ add info

{{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.

== 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 ==
TODO: independence from Peano (Legion) (see [[Logical independence]])

TODO: prurq new fast simulation method (see [https://discord.com/channels/960643023006490684/1471178503235043493 Discord thread])

TODO: "Cascade" (see [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 Discord thread] )

== 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 ==
*[[BB(2,5)]]: '''2 solved machines.'''
**Andrew Ducharme found a machine nonhalting on [https://discord.com/channels/960643023006490684/1259770421046411285/1471227102844944510 11 Feb] via the newly released mxdys FAR decider. This was verified in Rocq by mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471228798505582602 the same day].
**mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471229409829847111 announced another TM proven the same day], which turned out to be a translated cycler.
**Peacemaker II [https://discord.com/channels/960643023006490684/1259770421046411285/1472647706835943596 found the high-level behaviour of a machine], which turned out to be a relatively simple-to-describe string rewriting problem of sorts.
*[[BB(6)]]: '''XX''' machines simulated to 1e13, '''XX''' solved machines. TODO: Update
**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 newly released mxdys FAR decider.
**Alistaire [https://discord.com/channels/960643023006490684/1239205785913790465/1472376779825090713 found a machine nonhalting] using Quick_Sim.py.
**prurq simulated 38 machines for >1e13 steps[https://discord.com/channels/960643023006490684/1471178503235043493/1471486886890704967 <nowiki>[19 machines]</nowiki>][https://docs.google.com/spreadsheets/d/1zMhtW_edMxrfUry-hVMFsDg3T1p_udjC2V2RKu6oSKE/edit?usp=sharing <nowiki>[19 more machines]</nowiki>] with his new method [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 "Cascade".]
**Alistaire [https://discord.com/channels/960643023006490684/1239205785913790465/1472246267345113158 simulated 13 machines] for >1e13 steps, 6 of which had already been simulated by prurq, essentialy double-verifying them.
**Discord user @mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1472325414344069271 simulated a TM] for >1e13 steps, which also turned out to have been simulated by prurq already.
**Afterall, the informal holdout count is '''1299''', and the formal holdout count is 1303 (8 newly solved machines were solved by highly trusted code, 4 informal TMs beforehand). Rocq-verified holdout count is 1314. There remain '''160''' machines to be simulated up to 1e13. TODO: Update
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1473950417275850804 released] a [[holdouts list]] of 1226 machines up to equivalence, some of which were decided via [https://discord.com/channels/960643023006490684/1226543091264126976/1469937272752177298 new mxdys method for longitudinal acceleration].
**Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1474302212284092436 found 2 non-halting machines] in that list using the mxdys FAR decider.
***Another [https://discord.com/channels/960643023006490684/1239205785913790465/1475911180965904577 4 were found nonhalting] using the decider
*[[BB(7)]]:
**Andrew Ducharme has reduced the number of holdouts from 19,303,801 to 18,254,545 (a 5.44% reduction) and then '''18,195,192''' (0.33%) using the newly released mxdys 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 FAR decider.
***Another 0.47% reduction left 548,993 holdouts.[https://discord.com/channels/960643023006490684/1084047886494470185/1475216024734269644]

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

TMBR: February 2026

2026-02-23T03:36:41Z

Qwertyasdf: /* Holdouts */ add info

{{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.

== 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 ==
TODO: independence from Peano (Legion) (see [[Logical independence]])

TODO: prurq new fast simulation method (see [https://discord.com/channels/960643023006490684/1471178503235043493 Discord thread])

TODO: "Cascade" (see [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 Discord thread] )

== 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 ==
*[[BB(2,5)]]: '''2 solved machines.'''
**Andrew Ducharme found a machine nonhalting on [https://discord.com/channels/960643023006490684/1259770421046411285/1471227102844944510 11 Feb] via the newly released mxdys FAR decider. This was verified in Rocq by mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471228798505582602 the same day].
**mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471229409829847111 announced another TM proven the same day], which turned out to be a translated cycler.
**Peacemaker II [https://discord.com/channels/960643023006490684/1259770421046411285/1472647706835943596 found the high-level behaviour of a machine], which turned out to be a relatively simple-to-describe string rewriting problem of sorts.
*[[BB(6)]]: '''XX''' machines simulated to 1e13, '''XX''' solved machines. TODO: Update
**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 newly released mxdys FAR decider.
**Alistaire [https://discord.com/channels/960643023006490684/1239205785913790465/1472376779825090713 found a machine nonhalting] using Quick_Sim.py.
**prurq simulated 38 machines for >1e13 steps[https://discord.com/channels/960643023006490684/1471178503235043493/1471486886890704967 <nowiki>[19 machines]</nowiki>][https://docs.google.com/spreadsheets/d/1zMhtW_edMxrfUry-hVMFsDg3T1p_udjC2V2RKu6oSKE/edit?usp=sharing <nowiki>[19 more machines]</nowiki>] with his new method [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 "Cascade".]
**Alistaire [https://discord.com/channels/960643023006490684/1239205785913790465/1472246267345113158 simulated 13 machines] for >1e13 steps, 6 of which had already been simulated by prurq, essentialy double-verifying them.
**Discord user @mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1472325414344069271 simulated a TM] for >1e13 steps, which also turned out to have been simulated by prurq already.
**Afterall, the informal holdout count is '''1299''', and the formal holdout count is 1303 (8 newly solved machines were solved by highly trusted code, 4 informal TMs beforehand). Rocq-verified holdout count is 1314. There remain '''160''' machines to be simulated up to 1e13. TODO: Update
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1473950417275850804 released] a [[holdouts list]] of 1226 machines up to equivalence, some of which were decided via [https://discord.com/channels/960643023006490684/1226543091264126976/1469937272752177298 new mxdys method for longitudinal acceleration].
**Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1474302212284092436 found 2 non-halting machines] in that list using the mxdys FAR decider.
*[[BB(7)]]:
**Andrew Ducharme has reduced the number of holdouts from 19,303,801 to 18,254,545 (a 5.44% reduction) and then '''18,195,192''' (0.33%) using the newly released mxdys 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 FAR decider.
***Another 0.47% reduction left 548,993 holdouts.[https://discord.com/channels/960643023006490684/1084047886494470185/1475216024734269644]

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

TMBR: February 2026

2026-02-23T03:33:19Z

Qwertyasdf: updt

{{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.

== 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 ==
TODO: independence from Peano (Legion) (see [[Logical independence]])

TODO: prurq new fast simulation method (see [https://discord.com/channels/960643023006490684/1471178503235043493 Discord thread])

TODO: "Cascade" (see [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 Discord thread] )

== 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 ==
*[[BB(2,5)]]: '''2 solved machines.'''
**Andrew Ducharme found a machine nonhalting on [https://discord.com/channels/960643023006490684/1259770421046411285/1471227102844944510 11 Feb] via the newly released mxdys FAR decider. This was verified in Rocq by mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471228798505582602 the same day].
**mxdys [https://discord.com/channels/960643023006490684/1259770421046411285/1471229409829847111 announced another TM proven the same day], which turned out to be a translated cycler.
**Peacemaker II [https://discord.com/channels/960643023006490684/1259770421046411285/1472647706835943596 found the high-level behaviour of a machine], which turned out to be a relatively simple-to-describe string rewriting problem of sorts.
*[[BB(6)]]: '''XX''' machines simulated to 1e13, '''XX''' solved machines. TODO: Update
**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 newly released mxdys FAR decider.
**Alistaire [https://discord.com/channels/960643023006490684/1239205785913790465/1472376779825090713 found a machine nonhalting] using Quick_Sim.py.
**prurq simulated 38 machines for >1e13 steps[https://discord.com/channels/960643023006490684/1471178503235043493/1471486886890704967 <nowiki>[19 machines]</nowiki>][https://docs.google.com/spreadsheets/d/1zMhtW_edMxrfUry-hVMFsDg3T1p_udjC2V2RKu6oSKE/edit?usp=sharing <nowiki>[19 more machines]</nowiki>] with his new method [https://discord.com/channels/960643023006490684/1471178503235043493/1471178503235043493 "Cascade".]
**Alistaire [https://discord.com/channels/960643023006490684/1239205785913790465/1472246267345113158 simulated 13 machines] for >1e13 steps, 6 of which had already been simulated by prurq, essentialy double-verifying them.
**Discord user @mammillaria [https://discord.com/channels/960643023006490684/1239205785913790465/1472325414344069271 simulated a TM] for >1e13 steps, which also turned out to have been simulated by prurq already.
**Afterall, the informal holdout count is '''1299''', and the formal holdout count is 1303 (8 newly solved machines were solved by highly trusted code, 4 informal TMs beforehand). Rocq-verified holdout count is 1314. There remain '''160''' machines to be simulated up to 1e13. TODO: Update
**mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1473950417275850804 released] a [[holdouts list]] of 1226 machines up to equivalence,
**Andrew Ducharme [https://discord.com/channels/960643023006490684/1239205785913790465/1474302212284092436 found 2 non-halting machines] in that list using the mxdys FAR decider.
*[[BB(7)]]:
**Andrew Ducharme has reduced the number of holdouts from 19,303,801 to 18,254,545 (a 5.44% reduction) and then '''18,195,192''' (0.33%) using the newly released mxdys 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 FAR decider.
***Another 0.47% reduction left 548,993 holdouts.[https://discord.com/channels/960643023006490684/1084047886494470185/1475216024734269644]

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

Shift overflow counter

2026-02-19T10:41:17Z

Qwertyasdf: add common abbr

{{Stub}}
'''Shift overflow counter''' (SOC) is an informal class of Turing machines. A typical Turing machine in this class has the following behavior:

* It represents digits as short fixed-length blocks of symbols.
* It spends most of its time implementing basic double counter until one of the sides overflows (expands) which leads to changing the offsets of blocks, making them non-valid representations of digits.
* After “Counter Phase” there is a “Reset Phase” where the contents are “reparsed”, creating a new double counter configuration. The new configuration could lead to halting.

Note: some examples (like the halting shift-overflow counters below) use a counter on one side and a [[bouncer]] (sometimes called unary counter) on the other.

== Examples ==
* Skelet holdouts: [[Skelet 34]], [[Skelet 33]], [[Skelet 35]], [[Skelet 15]], [[Skelet 26]]
* {{TM|1RB0RF_1LC1RB_0RD0LB_---0LE_1RE0RA_1RD1RE}}

Halting shift-overflow counters:
* {{TM|1RB1LD_1RC0RB_1LA0LE_1LC0LA_1RZ1RB|halt}}
* {{TM|1RB1LD_1RC0RB_1LA1LE_1LC0LA_1RZ0RD|halt}}
* Current longest running [[BB(2,5)]] TM: {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
* Current seventh longest running [[BB(6)]] TM: {{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}

== Related links ==
* [https://www.sligocki.com/2023/02/02/skelet-34.html Skelet #34 is Infinite]
* [https://www.sligocki.com/2023/02/05/shift-overflow.html Shift Overflow Counters]
* [https://discuss.bbchallenge.org/t/skelet-26-and-15-do-not-halt-coq-proof/183 Skelet #26 and #15 do not halt - Coq proof]
* [https://discuss.bbchallenge.org/t/skelet-34-and-35-coq-proof/165 Skelet #34 and #35 – Coq proof]
* [https://discuss.bbchallenge.org/t/skelet-33-doesnt-halt-coq-proof/180 Skelet #33 doesn’t halt - Coq proof]

[[Category:Zoology]]

SOC

2026-02-19T10:40:39Z

Qwertyasdf: Redirected page to Shift overflow counter

#REDIRECT [[Shift overflow counter]]

BB(6)

2026-02-18T23:06:37Z

Qwertyasdf: /* Techniques */ clarify

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

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

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

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

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

All holdouts have been simulated up to at least 1e12 steps. 210 holdouts have not been simulated out to 1e13 steps. (The list can be found in the spreadsheet)
== Cryptids ==
Several [[Turing machines]] have been found that are [[Cryptids]], considered so because each of them have a [[Collatz-like]] halting problem, a type of problem that is generally difficult to solve. However, probabilistic arguments have allowed all but one of them to be categorized as [[probviously]] halting or probviously non-halting.

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

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

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|-
|{{TM|1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA|halt}}
|<math>> 10^{16\,780\,816}</math>
|Racheline
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

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

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

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

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

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

== References ==
<references />

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

BB(6)

2026-02-18T22:58:16Z

Qwertyasdf: not true Undo revision 6274 by BB fan weird (talk)

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

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

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

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

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

All holdouts have been simulated up to at least 1e12 steps. 210 holdouts have not been simulated out to 1e13 steps. (The list can be found in the spreadsheet)
== Cryptids ==
Several [[Turing machines]] have been found that are [[Cryptids]], considered so because each of them have a [[Collatz-like]] halting problem, a type of problem that is generally difficult to solve. However, probabilistic arguments have allowed all but one of them to be categorized as [[probviously]] halting or probviously non-halting.

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

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

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|-
|{{TM|1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA|halt}}
|<math>> 10^{16\,780\,816}</math>
|Racheline
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

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

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

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

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

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

== References ==
<references />

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

TMBR: February 2026

2026-02-14T16:58:01Z

Qwertyasdf: add info

BB(6)

2026-02-03T01:15:43Z

Qwertyasdf: /* Holdouts */ update

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

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

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

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

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

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

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

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

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

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

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

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
* {{TM|1RB0LD_0RC0RE_1LC0LA_---1LE_0LF0RE_0RA0LE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|-
|{{TM|1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA|halt}}
|<math>> 10^{11\,952\,340}</math> (lower bound)
|Racheline
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

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

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

All holdouts have been simulated up to at least 1e12 steps. 210 holdouts have not been simulated out to 1e13 steps. (The list can be found in the spreadsheet)

== References ==
<references />

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

TMBR: January 2026

2026-01-30T11:33:17Z

Qwertyasdf: /* Holdouts */ add solved 2,5 machine

{{TMBRnav|December 2025|February 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).''

This is the first edition of TMBR in 2026.

== 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.

== 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].

== 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)

== 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 ==

* [[BB(6)]]: '''2''' [Needs update] solved machines.
** Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire: see [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 spreadsheet]. Current count: '''210'''. (Total reduction: '''68'''. 38 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>] simulated out, plus 2 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].
** @Peacemaker II [https://discord.com/channels/960643023006490684/1239205785913790465/1466337165305974806 found] a machine 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
*[[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.
*[[BB(2,6)]]:
**Andrew Ducharme reduced the number of holdouts from 870,085 to '''869,001''', a 0.12% reduction.
**Another reduction brought the holdout count down to '''567,971'''.[https://discord.com/channels/960643023006490684/1084047886494470185/1465629456021323820]

[[Category:This Month in Beaver Research|2025-12]]

TMBR: January 2026

2026-01-30T11:20:03Z

Qwertyasdf: /* Holdouts */ add peacemaker result

{{TMBRnav|December 2025|February 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).''

This is the first edition of TMBR in 2026.

== 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.

== 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].

== 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)

== 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 ==

* [[BB(6)]]: '''2''' [Needs update] solved machines.
** Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire: see [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 spreadsheet]. Current count: '''210'''. (Total reduction: '''68'''. 38 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>] simulated out, plus 2 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].
** @Peacemaker II [https://discord.com/channels/960643023006490684/1239205785913790465/1466337165305974806 found] a machine 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
*[[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(2,6)]]:
**Andrew Ducharme reduced the number of holdouts from 870,085 to '''869,001''', a 0.12% reduction.
**Another reduction brought the holdout count down to '''567,971'''.[https://discord.com/channels/960643023006490684/1084047886494470185/1465629456021323820]

[[Category:This Month in Beaver Research|2025-12]]

Holdouts lists

2026-01-30T11:15:44Z

Qwertyasdf: update as of january

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide. Contributors have shared lists of holdouts. Some lists exclude known equivalent machines, i.e., machines whose halting status is implied by the halting status of another machine. If a holdout machine with more than one undefined transitions is found to reach a undefined transition, the number of holdouts may increase as a consequence of [[Tree Normal Form|TNF enumeration]].

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1314
|19,303,801
|-
!3-symbol
|0
|4
|9,401,447
|
|
|
|-
!4-symbol
|0
|12,435,284
|
|
|
|
|-
!5-symbol
|75
|
|
|
|
|
|-
!6-symbol
|567,971
|
|
|
|
|
|}

== Downloadable Holdout Lists ==
This is a table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages.
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(6)]]
|January 29th, 2026
|@mxdys
|1314
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 BB6_holdouts_1314.txt]
|
|-
|[[BB(6)]]
|December 30th, 2025
|@mxdys
|1326
|[https://wiki.bbchallenge.org/w/images/f/f0/BB6_holdouts_1326.txt BB6_holdouts_1326.txt]
|
|-
|[[BB(6)]]
|November 22nd, 2025
|Robin Rovenszky
|Variable (1325)
|[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Google Sheets]
|Annotated list, including links to Discord discussions, always up-to-date.
|-
|[[BB(6)]]
|December 16th, 2025
|@mxdys
|1343
|[https://wiki.bbchallenge.org/w/images/9/9a/BB6_holdouts_classified_1343.txt BB6_holdouts_1343.txt]
|
|-
|[[BB(6)]]
|November 28th, 2025
|@mxdys
|1416
|[https://wiki.bbchallenge.org/w/images/9/98/BB6_holdouts_1416.txt BB6_holdouts_1416.txt]
|
|-
|[[BB(6)]]
|November 14th, 2025
|@mxdys
|1534
|[https://wiki.bbchallenge.org/w/images/0/02/BB6_holdouts_1534.txt BB6_holdouts_1534.txt]
|
|-
|[[BB(6)]]
|October 20th, 2025
|@mxdys
|1618
|[https://wiki.bbchallenge.org/w/images/e/e3/BB6_holdouts_1618.txt BB6 holdouts 1618.txt]
|
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 22nd, 2025]
|@tjligocki
|873,469
|[http://2x6_holdouts_873469.txt.zip 2x6_holdouts_873469.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2,592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2,728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4,319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4,408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

Holdouts lists

2026-01-30T11:13:54Z

Qwertyasdf: /* Downloadable Holdout Lists */ add new holdouts list by mxdys

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide. Contributors have shared lists of holdouts. Some lists exclude known equivalent machines, i.e., machines whose halting status is implied by the halting status of another machine. If a holdout machine with more than one undefined transitions is found to reach a undefined transition, the number of holdouts may increase as a consequence of [[Tree Normal Form|TNF enumeration]].

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1326
|19,781,295
|-
!3-symbol
|0
|4
|9,401,447
|
|
|
|-
!4-symbol
|0
|12,435,284
|
|
|
|
|-
!5-symbol
|75
|
|
|
|
|
|-
!6-symbol
|869,001
|
|
|
|
|
|}

== Downloadable Holdout Lists ==
This is a table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages.
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(6)]]
|January 29th, 2026
|@mxdys
|1314
|[https://discord.com/channels/960643023006490684/1239205785913790465/1466438332677619956 BB6_holdouts_1314.txt]
|
|-
|[[BB(6)]]
|December 30th, 2025
|@mxdys
|1326
|[https://wiki.bbchallenge.org/w/images/f/f0/BB6_holdouts_1326.txt BB6_holdouts_1326.txt]
|
|-
|[[BB(6)]]
|November 22nd, 2025
|Robin Rovenszky
|Variable (1325)
|[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Google Sheets]
|Annotated list, including links to Discord discussions, always up-to-date.
|-
|[[BB(6)]]
|December 16th, 2025
|@mxdys
|1343
|[https://wiki.bbchallenge.org/w/images/9/9a/BB6_holdouts_classified_1343.txt BB6_holdouts_1343.txt]
|
|-
|[[BB(6)]]
|November 28th, 2025
|@mxdys
|1416
|[https://wiki.bbchallenge.org/w/images/9/98/BB6_holdouts_1416.txt BB6_holdouts_1416.txt]
|
|-
|[[BB(6)]]
|November 14th, 2025
|@mxdys
|1534
|[https://wiki.bbchallenge.org/w/images/0/02/BB6_holdouts_1534.txt BB6_holdouts_1534.txt]
|
|-
|[[BB(6)]]
|October 20th, 2025
|@mxdys
|1618
|[https://wiki.bbchallenge.org/w/images/e/e3/BB6_holdouts_1618.txt BB6 holdouts 1618.txt]
|
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 22nd, 2025]
|@tjligocki
|873,469
|[http://2x6_holdouts_873469.txt.zip 2x6_holdouts_873469.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2,592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2,728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4,319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4,408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

TMBR: January 2026

2026-01-30T11:12:21Z

Qwertyasdf: /* Holdouts */ add new holdouts list

{{TMBRnav|December 2025|February 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).''

This is the first edition of TMBR in 2026.

== 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.

== 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].

== 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)

== 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 ==

* [[BB(6)]]: '''2''' [Needs update] solved machines.
** Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire: see [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 spreadsheet]. Current count: '''210'''. (Total reduction: '''68'''. 38 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>] simulated out, plus 2 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].
** @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
*[[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(2,6)]]:
**Andrew Ducharme reduced the number of holdouts from 870,085 to '''869,001''', a 0.12% reduction.
**Another reduction brought the holdout count down to '''567,971'''.[https://discord.com/channels/960643023006490684/1084047886494470185/1465629456021323820]

[[Category:This Month in Beaver Research|2025-12]]

TMBR: January 2026

2026-01-27T12:47:14Z

Qwertyasdf: /* Holdouts */ update

{{TMBRnav|December 2025|February 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).''

This is the first edition of TMBR in 2026.

== 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.

== 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].

== 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)

== 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 ==

* [[BB(6)]]: '''2''' solved machines.
** Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire: see [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 spreadsheet]. Current count: '''210'''. (Total reduction: '''68'''. 38 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>] simulated out, plus 2 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].
*[[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(2,6)]]:
**Andrew Ducharme reduced the number of holdouts from 870,085 to '''869,001''', a 0.12% reduction.
**Another reduction brought the holdout count down to '''567,971'''.[https://discord.com/channels/960643023006490684/1084047886494470185/1465629456021323820]

[[Category:This Month in Beaver Research|2025-12]]

BB(6)

2026-01-21T08:28:32Z

Qwertyasdf: /* Techniques */ link

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

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

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

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

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

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

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

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

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

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

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

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
* {{TM|1RB0LD_0RC0RE_1LC0LA_---1LE_0LF0RE_0RA0LE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|-
|{{TM|1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA|halt}}
|<math>> 10^{11\,952\,340}</math> (lower bound)
|Racheline
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Holdouts ==
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
@mxdys's informal [[Holdouts lists|holdouts list]] has 1326 machines up to equivalence as of December 2025. Partial Rocq proof is [https://github.com/ccz181078/busycoq/tree/BB6 available on Github].

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

All holdouts have been simulated up to at least 1e12 steps. 210 holdouts have not been simulated out to 1e13 steps. (The list can be found in the spreadsheet)

== References ==
<references />

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

Holdouts lists

2026-01-14T08:14:55Z

Qwertyasdf: Editing prose for clarity; hope I explained TNF correctly

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide. Contributors have shared lists of holdouts. Some lists exclude known equivalent machines, i.e., machines whose halting status is implied by the halting status of another machine. If a holdout machine with more than one undefined transitions is found to reach a undefined transition, the number of holdouts may increase as a consequence of [[Tree Normal Form|TNF enumeration]].

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1326
|20,197,978
|-
!3-symbol
|0
|4
|9,401,447
|
|
|
|-
!4-symbol
|0
|12,435,284
|
|
|
|
|-
!5-symbol
|75
|
|
|
|
|
|-
!6-symbol
|870,085
|
|
|
|
|
|}

== Downloadable Holdout Lists ==
This is a table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages.
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(6)]]
|December 30th, 2025
|@mxdys
|1326
|[https://wiki.bbchallenge.org/w/images/f/f0/BB6_holdouts_1326.txt BB6_holdouts_1326.txt]
|
|-
|[[BB(6)]]
|November 22nd, 2025
|Robin Rovenszky
|Variable (1325)
|[https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=1330361301#gid=1330361301 Google Sheets]
|Annotated list, including links to Discord discussions, always up-to-date.
|-
|[[BB(6)]]
|December 16th, 2025
|@mxdys
|1343
|[https://wiki.bbchallenge.org/w/images/9/9a/BB6_holdouts_classified_1343.txt BB6_holdouts_1343.txt]
|
|-
|[[BB(6)]]
|November 28th, 2025
|@mxdys
|1416
|[https://wiki.bbchallenge.org/w/images/9/98/BB6_holdouts_1416.txt BB6_holdouts_1416.txt]
|
|-
|[[BB(6)]]
|November 14th, 2025
|@mxdys
|1534
|[https://wiki.bbchallenge.org/w/images/0/02/BB6_holdouts_1534.txt BB6_holdouts_1534.txt]
|
|-
|[[BB(6)]]
|October 20th, 2025
|@mxdys
|1618
|[https://wiki.bbchallenge.org/w/images/e/e3/BB6_holdouts_1618.txt BB6 holdouts 1618.txt]
|
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 22nd, 2025]
|@tjligocki
|873,469
|[http://2x6_holdouts_873469.txt.zip 2x6_holdouts_873469.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2,592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2,728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4,319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4,408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

TMBR: January 2026

2026-01-14T06:54:01Z

Qwertyasdf: add links

{{TMBRnav|December 2025|February 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).''

This is the first edition of TMBR in 2026.

== Champions ==

* [[User:Azerty|Azerty]] discovered a new [[BLB|BLB(3,3)]] [[champion]] ({{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}) which blanks the tape after running for more than 1042,745 steps.
* Azerty 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.

== 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(3)=17)

== 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 ==

* [[BB(6)]]: '''2''' solved machines.
** Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire: see [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 spreadsheet]. Current count: '''210'''. (Total reduction: '''68'''. 38 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>] simulated out, plus 2 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].
*[[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.[https://discord.com/channels/960643023006490684/1369339127652159509/1460718041577951365]

[[Category:This Month in Beaver Research|2025-12]]

TMBR: January 2026

2026-01-14T06:51:27Z

Qwertyasdf: add ConePine's proof idea

{{TMBRnav|December 2025|February 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).''

This is the first edition of TMBR in 2026.

== Champions ==

* [[User:Azerty|Azerty]] discovered a new [[BLB|BLB(3,3)]] [[champion]] ({{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}) which blanks the tape after running for more than 1042,745 steps.
* Azerty 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.

== 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(3)=17)

== 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 ==

* [[BB(6)]]: '''2''' solved machines.
** Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire: see [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 spreadsheet]. Current count: '''210'''. (Total reduction: '''68'''. 38 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>] simulated out, plus 2 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].
*[[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.
**The holdout count was further reduced to '''19,879,953''', a 1.57% reduction.

[[Category:This Month in Beaver Research|2025-12]]

TMBR: January 2026

2026-01-14T06:48:49Z

Qwertyasdf: grammar

{{TMBRnav|December 2025|February 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).''

This is the first edition of TMBR in 2026.

== Champions ==

* [[User:Azerty|Azerty]] discovered a new [[BLB|BLB(3,3)]] [[champion]] ({{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}) which blanks the tape after running for more than 1042,745 steps.
* Azerty 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.

== 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(3)=17)

== Holdouts ==

* [[BB(6)]]: '''2''' solved machines.
** Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire: see [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 spreadsheet]. Current count: '''210'''. (Total reduction: '''68'''. 38 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>] simulated out, plus 2 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].
*[[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.
**The holdout count was further reduced to '''19,879,953''', a 1.57% reduction.

[[Category:This Month in Beaver Research|2025-12]]

TMBR: January 2026

2026-01-14T01:09:26Z

Qwertyasdf: further reduction

{{TMBRnav|December 2025|February 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).''

This is the first edition of TMBR in 2026.

== Champions ==

* [[User:Azerty|Azerty]] discovered a new [[BLB|BLB(3,3)]] [[champion]] ({{TM|1RB2RB1LA_2LC0LB2LB_2RC2RA0LC}}) which blanks the tape after running for more than 1042,745 steps.
* Azerty 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.

== 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(3)=17)

== Holdouts ==

* [[BB(6)]]: '''2''' solved machines.
** Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire: see [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?gid=806905077#gid=806905077 spreadsheet]. Current count: '''210'''. (Total reduction: '''68'''. 38 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>] simulated out, plus 2 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].
*[[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.
**The holdout was further reduced to '''19,879,953''', a 1.57% reduction.

[[Category:This Month in Beaver Research|2025-12]]

BB(6)

2025-12-30T15:47:38Z

Qwertyasdf: wrong spelling

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

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

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

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

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

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

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

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

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

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

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

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
* {{TM|1RB0LD_0RC0RE_1LC0LA_---1LE_0LF0RE_0RA0LE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|-
|{{TM|1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA|halt}}
|<math>> 10^{11\,952\,340}</math> (lower bound)
|Racheline
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Holdouts ==
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
@mxdys's informal [[Holdouts lists|holdouts list]] has 1326 machines up to equivalence as of December 2025. Partial Rocq proof is [https://github.com/ccz181078/busycoq/tree/BB6 available on Github].

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

There are 12 holdouts not simulated up to 1e12 steps, and 287 holdouts not simulated up to 1e13 steps. (Lists can be found in the spreadsheet)

== References ==
<references />

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

BB(6)

2025-12-30T15:47:03Z

Qwertyasdf: add github link

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

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

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

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

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

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

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

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

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

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

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

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
* {{TM|1RB0LD_0RC0RE_1LC0LA_---1LE_0LF0RE_0RA0LE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
!Discoverer
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|mxdys
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|mxdys
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|Pavel Kropitz
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|poppuncher
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|poppuncher
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|Shawn Ligocki
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|Pavel Kropitz
|-
|{{TM|1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA|halt}}
|<math>> 10^{11\,952\,340}</math> (lower bound)
|Racheline
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Holdouts ==
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
@mxdys's informal [[Holdouts lists|holdouts list]] has 1326 machines up to equivalence as of December 2025. Partial Rcoq proof is [https://github.com/ccz181078/busycoq/tree/BB6 available on Github].

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

There are 12 holdouts not simulated up to 1e12 steps, and 287 holdouts not simulated up to 1e13 steps. (Lists can be found in the spreadsheet)

== References ==
<references />

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

BB(6)

2025-12-20T02:29:26Z

Qwertyasdf: /* Holdouts */ date update

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

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

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

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

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

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

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

@Iijil's [[MITMWFAR|MITMWFAR decider]] is likely too weak to be of any assistance: running the decider on 2650 BB(6) holdouts, using parameters not strong enough to solve BB(5) TMs, took prohibitively long to compute. Instead, [https://discord.com/channels/960643023006490684/1028746861395316776/1442964185599447152 a new FAR method] by mxdys was able to decide 88 of the 1534 holdouts.

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

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

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

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
* {{TM|1RB0LD_0RC0RE_1LC0LA_---1LE_0LF0RE_0RA0LE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|-
|{{TM|1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA|halt}}
|<math>> 10^{11\,952\,340}</math> (lower bound)
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Holdouts ==
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
@mxdys's informal [[Holdouts lists|holdouts list]] has 1343 machines up to equivalence as of December 2025.

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

There are 14 holdouts not simulated up to 1e12 steps, and 312 holdouts not simulated up to 1e13 steps. (Lists can be found in the spreadsheet)

== References ==
<references />

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

TMBR: December 2025

2025-12-16T02:22:33Z

Qwertyasdf: /* Holdouts */

{{TMBRnav|November 2025|January 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).''

This is the last edition of TMBR this 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.

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

=== Holdouts Reductions. ===

* [[BB(6)]] - Reduced from '''3571''' to '''1416''' holdouts. Hence, 2155 machines were solved this year. This is a '''60% reduction.'''
* [[BB(2,5)]] - Reduced from '''217''' to '''75,''' a '''65.43% reduction.'''
* [[BB(7)]] - '''Enumeration was completed''', the number of holdouts was reduced from an initial 85,853,789 to '''20,405,295''' machines, a 76.23% 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 14,518,243 holdouts, a '''96.66% reduction.'''
* [[BB(2,7)]] - '''Enumeration started''', 50K of the 1M subtasks have been enumerated ('''5%''').

=== 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)]] - [[User:Polygon|Polygon]] identified a new [[BB(4,3)]] champion with a score of over <math>10 \uparrow^{4} 4</math> ({{TM|1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD|halt}}). This TM was first proven to halt by Pavel Kropitz in May 2024, but its runtime was not known at the time.
=== 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]]s (Relative Movement Turing Machines) were introduced.
* 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(19) = 370 and exhaustive enumeration has been run up to size 21 (with BBf(21) ≥ 31,957,632 and 587 holdouts).
* [[Cyclic Tree Busy Beaver]] (CTBB) was introduced by @Jack on 14 Nov.[https://discord.com/channels/960643023006490684/1438694294042181742] 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).
TODO: Before July

=== 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).
* 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).
* 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>]
TODO: Before July

==BB Adjacent==
TODO. [[Register machine|Register machines]], [[General Recursive Function|General Recursive Functions]], Fractran progress.

== [[General Recursive Function|Holdouts]] ==
* [[BB(6)|BB(6):]]
**There are 14 holdouts left to simulate up to 1e12 steps, and 312 to simulate up to 1e13 steps[https://discord.com/channels/960643023006490684/1239205785913790465/1447303829400846482 <nowiki>[1]</nowiki>]. The two lists can be found [https://docs.google.com/spreadsheets/d/10y2aMp9yZVLyN-nXAxgA25rw4EA_lJqn3Vxy2kB-m-U/edit?gid=806905077#gid=806905077 here].
**The possibility of simulating computationally tractable machines which nonetheless has large time and memory requirements [https://discord.com/channels/960643023006490684/1448725136340422717 was discussed]. [[User:RobinCodes/Machines at the Edge#1RB0RE 1LC1LD 0RA0LD 1LB0LA 1RF1RA ---1LB (bbch) CRYPTID|List]]
** [[User:XnoobSpeakable|XnoobSpeakable]] continued reducing the number of holdouts with Stage 8 of Phase 2, by reducing it from 15,136,283 to 14,518,243 TMs. This is a 4.08% reduction.
* [[BB(2,7)|BB(2,7):]]
** Terry Ligocki enumerated 10K more subtasks, increasing the number of holdouts to 150,662,006 and making 50K of the 1 million subtasks enumerated or 5%.

[[Category:This Month in Beaver Research|2025-12]]

1RB1RE 1LC0RA 0RD1LB ---1RC 1LF1RE 0LB0LE

2025-12-12T13:34:48Z

Qwertyasdf: header added

{{machine|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}{{unsolved|Does this TM run forever?}}
{{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}, called '''[[Beaver Math Olympiad|BMO]] #1''', is a [[holdout]] [[BB(6)]] TM discovered by @-d on 25 Jun 2024 which is modeled by the [[Piecewise Affine Function]]:

<pre>
A(a, b) -> A(a-b, 4b+2) if a > b
A(a, b) -> A(2a+1, b-a) if a Halt if a = b

Start A(1, 2)
</pre>

for <code>A(a, b) = 0^inf 10^{a-1} 0 1^b E> 0^inf</code>. The configuration is always valid because <math>a \ge 1</math> is always maintained. Forward simulations of this model by @-d and Andrew Ducharme do not halt after 100,000,000 iterations of the map <code>A(a,b)</code>. It is the inspiration for the [[Beaver Math Olympiad]].

== Relation to Other Math Problems ==
This TM is a possible [[Cryptid]] since it seems hard to predict whether we could ever end up in <code>A(n, n)</code>. However, it is not (yet) classified as a Cryptid because there are no arguments for why this TM is hard (e.g., by comparing it to an open/unsolved/hard mathematical problem).

The map <code>A(., .)</code> is related to the study of discontinuous [https://en.wikipedia.org/wiki/Dynamical_system dynamical systems]. When +2 and +1 are ignored, the value of <math>\ln(a/b)</math> follows this chaotic map:

<math display="block">F(x) = \begin{cases}
\ln\left(\frac{e^x-1}{4}\right) & \text{if }x > 0 \\
\ln\left(\frac{2}{e^{-x}-1}\right) & \text{if }x < 0
\end{cases}</math>

Another potential BB(6) cryptid {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}} similarly iterates upon a tuple (x,y) with one function if x < y, another function if x > y, and halts if x = y.

This Turing Machine models a [[Piecewise Affine Function]], a class of iterated functions that are known to be Turing complete in general (similar to [[Generalized Collatz Problems]]).

== A Backward Reasoning Approach ==
An alternative approach is to work backwards and study the points which A(a,b) maps to the line a = b. For example, the points (2,7), (3,25), (4,71), and (5,181) all eventually map to a = b. (2,7) is immediately sent to a = b because, under the a A(2, 7) = (4 + 1, 7 - 2) = (5,5)</math>(5,181) requires more steps:

<math display="block">(5,181) \rightarrow (11, 176) \rightarrow (23, 165) \rightarrow (47, 142) \rightarrow (95, 95)</math>How are these points found? It's faster not to consider individual points, but as Racheline identified, the set of lines which are mapped to a = b. Introducing an index <math>n</math>, we can rewrite A(a,b) as<math display="block">(x_{n+1}, y_{n+1}) = \begin{cases} (x_n - y_n, 4y_n + 2), & \text{if } x_n \geq y_n \\ (2x_n + 1, y_n - x_n), & \text{if } x_n < y_n \end{cases}</math> If <math>x_{n+1} = y_{n+1}</math>{{machine|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE}}, then the lines with slopes <math>y_{n+1} = m x_{n+1} + b</math> map to that line. The two possible slopes are, for <math>x_n > y_n</math>and <math>x_n < y_n</math> respectively,

<math>4y_n + 2 = m (x_n - y_n) + b \iff y_n = \frac{m}{m + 4} x_n + \frac{b - 2}{m + 4}

</math> and<math>y_n - x_n = m (2 x_n + 1) + b \iff y_n = (2m + 1) x_n + (b + m).</math>

Indexing m and b in (slope, y-intercept) space gives

<math>(m_{k-1}, b_{k-1}) = \begin{cases}\left( \frac{m_k}{m_k + 4}, \frac{b_k - 2}{m_k + 4} \right), &x_{k-1} > y_{k-1} \\ (2m_k + 1, b_k + m_k), & x_{k-1} < y_{k-1} \end{cases}</math>.

Iteratively applying this (m,b) map finds an infinite set of points which A sends to a = b. The starting point is said line, with (m, b) = (1,0). Then the two lines of points which map to points on the line (m,b) = (1,0) under a single application A are have slopes and y-intercepts of <math>\left(\frac{1}{5}, -\frac{2}{5} \right) = \left(\frac{1}{1 + 4}, \frac{0-2}{1+4} \right) </math> and (3,1) = (2 + 1, 0 + 1). The four lines of points which map to (1,0) under two applications of A have slopes and y-intercepts <math>\left(\frac{1}{21}, -\frac{4}{7} \right), \left(\frac{7}{5}, -\frac{1}{5} \right), \left(\frac{3}{7}, -\frac{1}{7} \right)</math> and (7,4).
[[File:Slope intercept space after 5 iterations.png|thumb|Each point in the plotted (slope, intercept) space represents a line of points in (x,y) space that are eventually mapped to y = x by iterated application of the operation A. These points require five or fewer iterations to do so.]]
[[File:(m,b) space after 9 iterations with halting condition line.png|thumb|Each plotted point in (slope, intercept) space represents a line of points in (x,y) space which is mapped to y = x by nine or fewer iterated applications of A. If any point is on the black line, the TM in question halts.]]
To apply this to the problem at hand, we want to know if any of these generated lines contain the point (1,2), our TM-derived initial condition. The set of all lines (in position space, or (x,y) space) containing (1,2) are described by the line in (m,b) space 2 = m + b. If any of the generated points in (m,b) space sit on 2 = m + b, then the point (x,y) = (1,2) is eventually mapped by A to a point where x = y. In the second image, of (m,b) points after nine iterations, we see there are three possible locations for this to occur.
[[File:MbSpace iterations10 smallmb wline.png|center|thumb|575x575px|Each plotted point in (slope, intercept) space represents a line of points in (x,y) space which is mapped to y = x by ten or fewer iterated applications of A, with m > 2 and b > 0. If any point is on the black line, the TM in question halts. There are three possible locations. Two do not hit the line (in finite time). The third's status is unknown.]]

=== Location 1: Approaching (m = 1, b= 1) from below ===
There's two ways to see how (1,1) is not actually an intersection point. First, from Racheline, the elementary one: if m > b, m will always be greater than b, so a point where m = b is impossible.

Second, from Andrew, if one wants to generate a point near m = 1, you should create the largest possible value M to swamp out the additive +4 in the denominator of M/(M+4). M can be maximized with n iterations of the (m,b) map by only using the x < y operation. Then apply the x > y operation once for a near-unitary slope. The pair <math>(m_n, b_n)</math> formed by applying the x < y operation n times to (1,0) is <math>(2^{n+1}-1, 2^{n+1} -n-2)</math>. Apply the x > y operation to get

<math>\left( \frac{2^{n+1} - 1}{2^{n+1} + 3}, \frac{2^{n+1} -n-4}{2^{n+1}+3} \right)</math>.

Taking the limit as n goes to infinity, we get (1,1), so this TM does not reach the line 2 = m + b here in finite time.

=== Location 2: Approaching (m = 13/7, b = 4/21) from above ===
The rightmost possible intersection is the least likely: its approach towards 2 = m + b slows considerably with increasing iterations (because its limit will end up being to above this line). Empirically examining these points, they appear to be near (1.85, 0.19). After 6 through 9 iterations, the closest of the generated (m,b) points to (1.85, 0.19) are (285/151, 37/151), (1117/599, 123/599), (4445/2391, 155/797), and (17757/9559, 1831/9559). Such points are formed by n applications of the (m,b) map when x > y, then two applications of the (m,b) map when x < y, then a single additional application of the x > y and x < y (m,b) maps, in that order. A more concise notation would be <math>(>^n, <^2, >, <)</math>.

A closed form equation in n for <math>(>^n, <^2, >, <)</math> starting from (1,0) is <math>\left( \frac{13 * 4^{n+1} + 23}{7* 4^{n+1} + 5}, \frac{4^{n+2} + 6n + 65}{21 * 4^{n+1} + 15} \right)</math>. Its limit as <math>n \rightarrow \infty</math> is <math>\left( \frac{13}{7}, \frac{4}{21} \right)</math>, which is safely above 2 = m + b. This strand of points will never cause a halting condition.

Racheline showed the same by applying the bounds m > 0 and <math>b > - \frac{2}{3}</math> on the sequence <math>(<^2, >, <)</math>:

<math>\begin{align} m_0 > 0, b_0 > - \frac{2}{3} &\implies m_1 > 1, b_1 > -\frac{2}{3} \\
&\implies m_2 > 3, b_2 > \frac{1}{3} \\
&\implies m_3 > \frac{3}{7}, b_3 > -\frac{5}{21} \\
&\implies m_4 > \frac{13}{7}, b_4 > \frac{4}{21}
\end{align}</math>

Since the combination of operations is empirically determined, this isn't strictly a proof, but the moderate distance of these points from 2 = m + b means the burden of proof isn't that high.

=== Location 3: Jumping above and below b = 2 - m around m = 1.76...infinitely? ===
Notes from my (Andrew) analysis, from when I tried to repeat looking at the points closest to the not-quite-intersection of the sparse set and line b = 2 - m. I went 25 iterations of the slope map before realizing it was likely a fractal and zooming in would give me more of the same:
Slopes of points (m,b) closest to the line m = 2 - b from above and below

13/7 for n = 5 is above and to the right, 119/69 for n = 6 is below and to the left, 223/125 for n = 7 is above and to the right, 669/401 for n = 8 is below left.
Things start to get more complicated at n = 9, as the doubling of dots becomes a little overwhelming.
So many dots at n = 13 you have to zoom in
n = 9 above: 223/125, n = 9 below: 1145/651
n = 10 above: 2037/1151, n = 10 below: 1145/651 (still)
n = 11 above: 6617/3755, n = 11 below: 1145/651 (still)
n = 12 above: 6617/3755 (still), n = 12 below: 24937/14171
n = 13 above: 6617/3755 (still), n = 13 below: 38171/21681
n = 14 above: 6617/3755 (still), n = 14 below: 64639/36701
n = 15 above: 117575/66741, n = 15 below: 64639/36701 (still)
n = 16 above: 117575/66741 (still), n = 16 below: 376131/213545
n = 17 above: 611281/347027, n = 17 below: 376131/213545 (still).
b for the new point here is 2 / 347027 from being on the line!!
At n = 17, have we crossed the Rubicon? This super close point is just above the line, so is anything new now above it?
No! yowza.
n = 18 above: 611281/347027 (still), n = 18 below: 611281/347027.
n = 19 above: 611281/347027 (still), n = 19 below: 2115805/1201207
n = 20 above: 611281/347027 (still), n = 20 below: 3338367/1895261
n = 21 above:611281/347027 (still) , n = 21 below: 10673739/6059585
There's a new point next to our hero which is just to the right and just above. Above trend hitting limit point?
NO!!
n = 22 above: 208917121/118603171, n = 22 below: 20454235/11612017 I might be seeing a new trend line forming
n = 23 above: 834868865/473958819, n = 23 below: 40015227/22716881
n = 24 above: 3338675841/1895381411, n = 24 below: 40015227/22716881 (still)
n = 25 above: 79137211/44926609, n = 25 below: 239198119/135794133

I think I'm zooming in on a fractal structure and will never see the end of this.
[[File:Screenshot of (m,b) set as we zoom in on possible intersection.png|center|thumb|636x636px|To demonstrate the fractal, the generated (m,b) set after n iterations for n = 9, n = 15, n = 20, and n = 25. Note the change in scale on the horizontal and vertical axes. I'm roughly specifying another decimal place in the axes bounds as I look at smaller and smaller region. Largely a copy of post @aducharme [https://discord.com/channels/960643023006490684/1026577255754903572/1267352007811797012 July 27th, 2024 10:24 PM PT]]]
Unless we actually hit the halting condition, we can generate an infinite number of blue points arbitrarily close to the yellow line from both above and below.

It does appear that points within roughly the range 1.724 < m < 1.8556 are generated from finding all possible combinations of operations > and < after n applications, then applying <math>(<,<,<,>,<,>,<) = (<^4, (>,<)^2) = (<^4, >,<,>,<)</math>. This is equivalent to <math display="block">(m_f,b_f) = \left( \frac{104m_n + 119}{56 m_n + 69}, \frac{23m_n + b_n + 12}{56m_n + 69} \right).</math>

== Study of Simplified Model ==
@savask [https://discord.com/channels/960643023006490684/1026577255754903572/1255778174986485780 06/26/2024 11:54 PM]
Maybe we could tackle a simplified version of this problem? For example, let's remove the constant terms:

(a, b) -> (a-b, 4*b) if a > b
(a, b) -> (2*a, b-a) if b > a
(a, b) -> Halt if a = b

And let's start from config (3,17), since it appears in the run of the original machine.
The average increase in <math>\ln a</math> is

<math>\frac{-\frac{\pi^2}{12} + 3 \frac{(\ln 2)^2 }{2} + \text{Li}_2(1/4)}{\ln 3} \approx 0.15097694</math>

when modeled from the PDF

<math>f(r) = \frac{2}{\ln 3 (r+1)(r+3)}</math>

in <math>r = b/a</math>, as found by [https://discord.com/channels/960643023006490684/1026577255754903572/1256315391995809842 @LegionMammal978] and Math.SE user "Sil".<ref>https://math.stackexchange.com/questions/4939082/solving-the-functional-equation-fx-frac4x42f-frac-xx42f2x1/4939136#4939136</ref>

Note that the simplified iteration trivially cannot halt as written: after 2 steps, it reaches <math>(12, 8)</math> and becomes trapped within the families <math>(a, b) \equiv (0, 4)</math> and <math>(a, b) \equiv (4, 0) \pmod{8}</math>.

== Development of Model from TM Tape ==
@-d [https://discord.com/channels/960643023006490684/1026577255754903572/1255047256688824390 25 Jun 2024 2:29 AM ET]:
<pre>
Let f(a,b) represent (10)^a D> 1^b. By inspection, the machine visits f(0,3) f(2,2) f(1,7) f(4,5) f(0,19) f(2,18) f(6,15) f(14,8).

I got these rules for f(a,b) and verified them up to 0<=a<=100 and 1<=b<=100.
f(a,b) -> f(a-b+1,4b-1) if b < a+2
f(a,a+2) -> halt
f(a,b) -> f(2a+2,b-a-1) if b > a+2

I ran these rules from f(0,3) and gave up after reaching a,b > 2^1000000. It looks like halting becomes less and less likely. Is there a way to show this runs forever (or miraculously halts)?
</pre>

@Shawn Ligocki — [https://discord.com/channels/960643023006490684/1026577255754903572/1255371132694167591 25 Jun 2024 at 11:56 PM ET]:
<pre>
I can confirm roughly the same rules (I chose a slightly different standard config):
1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE

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

Base Rules:
E(a+1, b, c+2) -> E(a, b+2, c+1)
E(a+1, b, 1) -> E(a, 0, 2b+6)
E(0, b, c+2) -> E(b+2, 0, c+1)
E(0, b, 1) -> Halt(b+4)

High-level Rules:
E(a, 0, c+1) --> E(a-c-1, 0, 4c+6) if a > c
E(a, 0, c+1) --> E(2a+2, 0, c-a) if a < c
E(a, 0, c+1) --> Halt(2c+4) if a = c

Connecting to your notation @-d : f(a, b+2) -> E(a, 0, b+1)
</pre>

@Shawn Ligocki — 26 Jun 2024 at 12:25 AM
<pre>
Hm, this is interesting. I decided to track "spread" akin to what @savask and I have been looking at for the similar 3x3 TM we're looking at in ⁠bbxy . Here I define spread = abs(a - c) in config E1(a, c) = E(a, 0, c+1). Here's the results I have so far:
1 E1(2, 0) spread: [2, 2] (0.06s)
100_001 E1(10^6_557, 10^6_555) spread: [1, 10^6_557] (0.44s)
200_001 E1(10^13_124, 10^13_125) spread: [10^6_557, 10^13_125] (1.27s)
300_001 E1(10^19_670, 10^19_669) spread: [10^13_123, 10^19_670] (2.52s)
400_001 E1(10^26_235, 10^26_237) spread: [10^19_668, 10^26_237] (4.23s)
500_001 E1(10^32_799, 10^32_799) spread: [10^26_235, 10^32_799] (6.39s)
600_001 E1(10^39_350, 10^39_352) spread: [10^32_799, 10^39_352] (8.98s)
700_001 E1(10^45_930, 10^45_930) spread: [10^39_350, 10^45_930] (12.06s)
800_001 E1(10^52_486, 10^52_486) spread: [10^45_929, 10^52_486] (15.54s)
900_001 E1(10^59_046, 10^59_047) spread: [10^52_485, 10^59_047] (19.57s)
1_000_001 E1(10^65_613, 10^65_612) spread: [10^59_045, 10^65_613] (24.05s)
1_100_001 E1(10^72_154, 10^72_153) spread: [10^65_609, 10^72_154] (28.92s)
1_200_001 E1(10^78_728, 10^78_729) spread: [10^72_153, 10^78_729] (34.21s)
1_300_001 E1(10^85_288, 10^85_289) spread: [10^78_729, 10^85_289] (40.03s)
1_400_001 E1(10^91_830, 10^91_829) spread: [10^85_288, 10^91_829] (46.40s)
1_500_001 E1(10^98_394, 10^98_393) spread: [10^91_829, 10^98_394] (53.11s)
1_600_001 E1(10^104_961, 10^104_959) spread: [10^98_393, 10^104_961] (60.18s)
1_700_001 E1(10^111_523, 10^111_523) spread: [10^104_958, 10^111_523] (67.75s)
1_800_001 E1(10^118_088, 10^118_088) spread: [10^111_523, 10^118_088] (75.77s)
1_900_001 E1(10^124_631, 10^124_630) spread: [10^118_088, 10^124_631] (84.39s)
2_000_001 E1(10^131_185, 10^131_185) spread: [10^124_630, 10^131_185] (93.41s)
...
3_000_001 E1(10^196_733, 10^196_733) spread: [10^190_163, 10^196_733] (217.87s)

Here spread: [X, Y] means that the spread values (since last print) had min X, max Y
So notice that the spread is basically uniformly increasing. In fact these [X, Y] intervals hardly even overlap!
That makes me think there's something about this that makes small spread impossible (and thus halt impossible)
Or that it's like Bigfoot and it ... could go back to zero, but the chances are infinitesimal!
</pre>

savask — 26 Jun 2024 at 1:02 AM
<pre>
Maybe we can simplify a little bit more. Set (a, c) = E(a, 0, c+1) then the rules are
(a, c) -> (a-c-1, 4c+5) if a > c
(a, c) -> (2a+2, c-a-1) if a < c
(a, c) -> Halt if a = c

I like that this way they are more symmetrical with a-c-1 vs c-a-1
</pre>

Shawn Ligocki — 26 Jun 2024 at 2:33 AM
<pre>
FWIW, there is a value x ~= 1.76 such that as c -> inf: (xc, c) -> (xyc, yc) via path "rll" (right->0 [ie a > c], left->0 [ie a < c], left->0). Which would repeat forever if it were exact, but it is an unstable cycle so any slight deviation and it gets further and further away, so that doesn't seem like a reasonable proof direction :/
</pre>dyuan01 — 26 Jun 2024 at 2:32 PM
If we let A(a, c) = (a-1, c-1), we can get these rules:
```
A(a, c) -> A(a-c, 4c+2) if a > c
A(a, c) -> A(2a+1, c-a) if a < c
A(a, c) -> Halt if a = c
```
And we start with (1, 2)

I'm not sure if it even helps, but at least the rules look slightly more "human-manageable"

== Analysis by Bard ==
Under a change of coordinates, the BMO1 rules approach a 1D line in the limit as the number of steps tends to infinity. Analysing the dynamics of points on that 1D line, it is possible to prove any given point on the line either halting or looping. Unfortunately, since at finite time, the point only gets arbitrarily close to the limiting line and never reaches it, it is not clear (to me at least) how to make use of this fact to solve BMO1. See the paper at [https://wiki.bbchallenge.org/w/images/9/94/BMO1-Bard.pdf BMO1 - Bard.pdf].

== References ==
[[Category:BB(6)]]

BB(6)

2025-12-04T13:02:50Z

Qwertyasdf: add info

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

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

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

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

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

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

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

@Iijil's [[MITMWFAR|MITMWFAR decider]] is likely too weak to be of any assistance: running the decider on 2650 BB(6) holdouts, using parameters not strong enough to solve BB(5) TMs, took prohibitively long to compute. Instead, [https://discord.com/channels/960643023006490684/1028746861395316776/1442964185599447152 a new FAR method] by mxdys was able to decide 88 of the 1534 holdouts.

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

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

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

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
* {{TM|1RB0LD_0RC0RE_1LC0LA_---1LE_0LF0RE_0RA0LE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|-
|{{TM|1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA|halt}}
|<math>> 10^{11\,952\,340}</math> (lower bound)
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Holdouts ==
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
@mxdys's informal [[Holdouts lists|holdouts list]] has 1416 machines up to equivalence as of November 2025.

Always up-to-date annotated spreadsheet, with links to Discord discussions: [https://docs.google.com/spreadsheets/d/10y2aMp9yZVLyN-nXAxgA25rw4EA_lJqn3Vxy2kB-m-U Spreadsheet]

== References ==
<references />

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

TMBR: November 2025

2025-12-01T23:22:16Z

Qwertyasdf:

{{TMBRnav|October 2025|December 2025}}

''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).''

== In the News ==
* Dr. Carl Kadie [https://discord.com/channels/960643023006490684/960643023530762343/1440090541936214017 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)'']

== Themed Months ==

There have been two specific BB domain themed months: BB(3,3) month (October) and BB(2,5) month (November). For BB(3,3), this resulted in the clarification 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]. LegionMammal wrote up [https://discord.com/channels/960643023006490684/1259770474897080380/1431274106736873685 a detailed explanation] of the probvious halter. No further progress on holdouts was made in October due to the limited number of holdouts and the work that's already been put into trying to solve them. For BB(2,5), it was verified that the [https://wiki.bbchallenge.org/wiki/BB(2,5) Wiki] was up to date. @mxdys shared some of [https://discord.com/channels/960643023006490684/1259770421046411285/1436149296004071615 his old notes] on the unanalyzed machines.

== Misc ==

* Discord user Vonhust made a [[Macro Machine]] achieving 2 billion steps / s, with more optimizations planned. TODO: Sources, details, [https://discord.com/channels/960643023006490684/1226543091264126976/1438890558499061821 graph].

==BB Adjacent==

* [[File:Fractran deciders.png|alt=All 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]]][[Fractran|Busy Beaver for Fractan, or BBf]] was introduced on 1 Nov by Jason Yuen, who also solved the first 14 values. Then, all values up to 19 were solved, with 19 requiring 3 programs to be proven by hand. For n = 20, 34 holdouts remain, while for n = 21, 587 holdouts are remaining. Three BB-Cryptids were also transformed into Fractran programs, so we know that n = 29 requires solving Hydra.
* [[Cyclic Tag]] was introduced by Discord user Jack on 14 Nov. The first two values, n = 1 and 2 were also solved by him. Then, lower bounds were given for values up to n = 7. CTBB(7) reaches hexation.
* TODO. Add BMS milestone improved by tromp from 350 to 331. (source: https://discord.com/channels/960643023006490684/1355653587824283678/1436844104742076507)
* TODO. Add [[Busy Beaver for SKI calculus]].

= Holdouts =

* [[BB(6)|BB(6):]] Total holdout reduction: 204 TMs, a 12% reduction!
** @mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1438983076662477004 released a holdouts list] consisting of 1534 machines up to equivalence, which is a 6% reduction (84 machines)
** @mxdys [https://discord.com/channels/960643023006490684/1400456788955893840/1438559730640355378 decided a holdout] from [https://discord.com/channels/960643023006490684/1400456788955893840 50 Random Holdouts] by hand, making 11/50 solved. He also [https://discord.com/channels/960643023006490684/1239205785913790465/1440413375950487693 decided 5 more machines] with a single strategy. (This is not a holdout in the 1534 list, just progress on the 50 Random holdouts).
** At the end of the month, @mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1443989662019354647 released a holdouts list] consisting of only 1416 holdout TMs. A total of 118 holdouts were solved. 88 [https://discord.com/channels/960643023006490684/1028746861395316776/1442964185599447152 by a new FAR method], created by mxdys. 30 more by 14 new individual proofs by mxdys and 16 informal proofs verified.
** [https://docs.google.com/spreadsheets/d/10y2aMp9yZVLyN-nXAxgA25rw4EA_lJqn3Vxy2kB-m-U/edit?gid=1330361301#gid=1330361301 A new holdouts list] was created by Robin Rovenszky via searching the Discord server for any results on TMs. This spreadsheet thus includes every informal result alongside verified results.
** Right after, [https://discord.com/channels/960643023006490684/1239205785913790465/1443997231970648206 Peacemaker decided a TM by hand.] @mxdys verified this result using an [[Inductive Proof System|Inductive]] decider.
** With substantial work, [https://discord.com/channels/960643023006490684/1443295684878143579/1443295684878143579 a machine was formally proven nonhalting] by Pomme, Autumn Pan, vyx and mxdys. However, this result is yet to be verified.
* [[BB(2,7)|BB(2,7):]] Enumeration started!
** The [https://github.com/ccz181078/TM code provided by mxdys] breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].
** Terry Ligocki enumerated the first 3 subtasks, resulting in 88,730,540 holdouts, which can be found [https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link here].
** The expected number of holdouts after the enumeration is ~3B TMs.

[[Category:This Month in Beaver Research|2025-11]]

BB(6)

2025-11-27T03:47:17Z

Qwertyasdf: link

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

The current BB(6) [[champion]] {{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

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

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

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

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

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

@Iijil's [[MITMWFAR|MITMWFAR decider]] is likely too weak to be of any assistance: running the decider on 2650 BB(6) holdouts, using parameters not strong enough to solve BB(5) TMs, took prohibitively long to compute.

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

Probviously non-halting Cryptids:

* {{TM|1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA}}, [[Antihydra]]
* {{TM|1RB1RC_1LC1LE_1RA1RD_0RF0RE_1LA0LB_---1RA|undecided}}, a variant of [[Hydra]] and Antihydra
* {{TM|1RB1LD_1RC1RE_0LA1LB_0LD1LC_1RF0RA_---0RC|undecided}}, similar to Antihydra
* {{TM|1RB0LD_1RC1RF_1LA0RA_0LA0LE_1LD1LA_0RB---|undecided}}, similar to Antihydra
* {{TM|1RB0LB_1LC0RE_1LA1LD_0LC---_0RB0RF_1RE1RB|undecided}}, similar to Antihydra
* {{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD|undecided}}

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
* {{TM|1RB0LD_1RC1RA_1LD0RB_1LE1LA_1RF0RC_---1RE}}
* {{TM|1RB1LD_1RC0LE_1LA1RE_0LF1LA_1RB0RB_---0LB}}
* {{TM|1RB0RE_1LC0RA_1LA1LD_1LC1LF_0LC0LB_1LE---}}

Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

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

Potential Cryptids:

* {{TM|1RB1RE_1LC0RA_0RD1LB_---1RC_1LF1RE_0LB0LE|undecided}}
* {{TM|1RB0LD_1LC0RA_1RA1LB_1LA1LE_1RF0LC_---0RE|undecided}}
* {{TM|1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC|undecided}}
* {{TM|1RB0LD_0RC0RE_1LC0LA_---1LE_0LF0RE_0RA0LE|undecided}}
* {{TM|1RB1RF_1LC1LF_0RE1LD_0LB1LD_---1RC_1RA0RD|undecided}}

== Top Halters ==
Below is a table of the machines with the 20 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]].<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
|-
|{{TM|1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}}
|2 ↑↑↑ 5
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE|halt}}
|10 ↑↑ 15.60465
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF|halt}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA|halt}}
|10 ↑↑ 7.23619
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB|halt}}
|10 ↑↑ 6.96745
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE|halt}}
|10 ↑↑ 5.77573
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD|halt}}
|10 ↑↑ 5.63534
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC|halt}}
|10 ↑↑ 5.56344
|-
|{{TM|1RB0LE_0RC1RA_0LD1RF_1RE0RB_1LA0LC_0RD1RZ|halt}}
|10 ↑↑ 5.12468
|-
|{{TM|1RB0RF_1LC1LB_0RE0LD_0LC0LB_0RA1RE_0RD1RZ|halt}}
|10 ↑↑ 5.03230
|-
|{{TM|1RB1LA_1LC0RF_1LD1LC_1LE0RE_0RB0LC_1RZ1RA|halt}}
|10 ↑↑ 4.91072
|-
|{{TM|1RB0LE_1LC1RA_1RE0LD_1LC1LF_1LA0RC_1RZ1LC|halt}}
|10 ↑↑ 3.33186
|-
|{{TM|1RB1RF_1LC1RE_0LD1LB_1LA0RA_0RA0RB_1RZ0RD|halt}}
|10 ↑↑ 3.31128
|-
|{{TM|1RB0LF_1LC0RA_1RD0LB_1LE1RC_1RZ1LA_1LA1LE|halt}}
|10 ↑↑ 3.18855
|-
|{{TM|1RB0RF_1LC1RB_0RD0LB_1RZ0LE_1RE0RA_1RD1RE|halt}}
|10 ↑↑ 3.16005
|-
|{{TM|1RB1RZ_0LC0LD_1LD1LC_1RE1LB_1RF1RD_0LD0RA|halt}}
|<math>10^{646\,456\,993}</math>
|-
|{{TM|1RB0RC_0LC0LB_0LD1LC_0LE1LA_0LF---_1RF1RA|halt}}
|<math>> 10^{11\,952\,340}</math> (lower bound)
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

== Holdouts ==
[[File:BB(6) holdouts decrease over time.png|alt=BB(6) Holdouts count decrease overtime.|thumb|Number of BB(6) holdouts over time.]]
@mxdys's informal [[Holdouts lists|holdouts list]] has 1534 machines up to equivalence as of November 2025.

Always up-to-date annotated spreadsheet, with links to Discord discussions: [https://docs.google.com/spreadsheets/d/10y2aMp9yZVLyN-nXAxgA25rw4EA_lJqn3Vxy2kB-m-U Spreadsheet]

== References ==
<references />

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

Holdouts lists

2025-11-22T16:13:19Z

Qwertyasdf: add Robin Code's holdout list

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide.

Holdout lists are often shared by contributors. There is a [[#Downloadable Holdout Lists|Downloadable Holdout Lists]] table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages. They may exclude known equivalent machines, whose halting status is implied by the halting status of another machine.

The table with the "Number of holdouts" is based on the holdout lists listed in the table below it. Thus, some of these numbers have not been independently verified. All the zero entries (no remaining holdouts) have been verified.

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1,534
|20,405,295
|-
!3-symbol
|0
|6
|9,401,447
|
|
|
|-
!4-symbol
|0
|15,136,283
|
|
|
|
|-
!5-symbol
|75
|
|
|
|
|
|-
!6-symbol
|870,085
|
|
|
|
|
|}

== Downloadable Holdout Lists ==
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(6)]]
|November 23th, 2025
|Robin Codes
|1534
|[https://docs.google.com/spreadsheets/d/10y2aMp9yZVLyN-nXAxgA25rw4EA_lJqn3Vxy2kB-m-U/edit?gid=1330361301#gid=1330361301 Google Sheets]
|Annotated list, including links to Discord discussions
|-
|[[BB(6)]]
|November 14th, 2025
|@mxdys
|1534
|[https://wiki.bbchallenge.org/w/images/0/02/BB6_holdouts_1534.txt BB6_holdouts_1534.txt]
|
|-
|[[BB(6)]]
|October 20th, 2025
|@mxdys
|1618
|[https://wiki.bbchallenge.org/w/images/e/e3/BB6_holdouts_1618.txt BB6 holdouts 1618.txt]
|
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 22nd, 2025]
|@tjligocki
|873,469
|[http://2x6_holdouts_873469.txt.zip 2x6_holdouts_873469.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2,592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2,728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4,319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4,408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

Holdout lists

2025-11-16T13:20:19Z

Qwertyasdf: Redirected page to Holdouts lists

#REDIRECT [[Holdouts lists]]

Holdouts lists

2025-10-17T06:10:05Z

Qwertyasdf:

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide.

Holdout lists are often shared by contributors. There is a [[#Downloadable Holdout Lists|Downloadable Holdout Lists]] table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages. They may exclude known equivalent machines, whose halting status is implied by the halting status of another machine.

The table with the "Number of holdouts" is based on the holdout lists listed in the table below it. Thus, some of these numbers have not been independently verified. All the zero entries (no remaining holdouts) have been verified.

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1,691
|22,721,168
|-
!3-symbol
|0
|6
|33,860,069
|
|
|
|-
!4-symbol
|0
|17,983,810
|
|
|
|
|-
!5-symbol
|75
|
|
|
|
|
|-
!6-symbol
|873,469
|
|
|
|
|
|}

== Downloadable Holdout Lists ==
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 22nd, 2025]
|@tjligocki
|873,469
|[http://2x6_holdouts_873469.txt.zip 2x6_holdouts_873469.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2,592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2,728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4,319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4,408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

Holdouts lists

2025-10-17T06:09:22Z

Qwertyasdf: clarify

A '''holdout''' (or undecided machine) is a [[Turing machine]] for which it is not known whether the machine halts or not from all-0 input tape. Holdouts are the machines which [[Decider|deciders]] are yet unable to decide.

Holdout lists are often shared by contributors. There is a [[#Downloadable Holdout Lists|Downloadable Holdout Lists]] table where people have added lists with no restriction or independent verification. For some of the entries there is a reference to a spreadsheet that documents what was run to achieve the result. For others, there is additional documentation on the specific BB pages. They may exclude equivalent machines, whose halting status is implied by the halting status of another machine.

The table with the "Number of holdouts" is based on the holdout lists listed in the table below it. Thus, some of these numbers have not been independently verified. All the zero entries (no remaining holdouts) have been verified.

{| class="wikitable"
|+Number of holdouts
!
!2-state
!3-state
!4-state
!5-state
!6-state
!7-state
|-
!2-symbol
|0
|0
|0
|0
|1,691
|22,721,168
|-
!3-symbol
|0
|6
|33,860,069
|
|
|
|-
!4-symbol
|0
|17,983,810
|
|
|
|
|-
!5-symbol
|75
|
|
|
|
|
|-
!6-symbol
|873,469
|
|
|
|
|
|}

== Downloadable Holdout Lists ==
{| class="wikitable sortable"
|+
!BB space
!Date
!Shared by
!Number of holdouts
!File
!Notes
|-
|[[BB(3,4)]]
|[https://drive.google.com/drive/folders/1_5j19qrvo1q7jN_c0pYnjBOrIXAP6b7i October 4th, 2025]
|@tjligocki
|64,777,377
|[https://drive.google.com/file/d/1I3s3w-T4NPLn-eCdaZLPgC2Mya51pyvH/view?usp=drive_link 3x4_holdouts_64777377.txt.gz]
|Work done by [[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]].
|-
|[[BB(7)]]
|[https://discord.com/channels/960643023006490684/1369339127652159509/1423806362072256676 October 4th, 2025]
|Andrew Ducharme
|22,721,168
|[https://drive.google.com/file/d/1xAFSPh6qAR8VxsF4QVipmdc0UpdPC1hn/view bb7_holdouts_22721168.txt.zip]
|
|-
|[[BB(2,6)]]
|[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT September 22nd, 2025]
|@tjligocki
|873,469
|[http://2x6_holdouts_873469.txt.zip 2x6_holdouts_873469.txt.zip]
|Work done by Andrew Ducharme.
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1411308408220549140 August 30 2025]
|@mxdys
|2,592
|[[:File:BB6 holdouts 2592.txt|BB6 holdouts 2592.txt]]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1399783936019664896 July 29 2025]
|@mxdys
|2,728
|[[:File:BB6 holdouts 2728.txt|BB6 holdouts 2728.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1259770421046411285/1355593937531961365 March 29 2025]
|@mxdys
|83
|[[:File:BB2x5 Coq holdouts 83.txt]]
|
|-
|[[BB(4,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1320172124509311004 December 21, 2024]
|@tjligocki
|460,916,384
|[https://drive.google.com/file/d/1hKy0TuPuI62rN95j6ZLjXgE-Pue8tRsK/view?usp=drive_link 4x3_holdouts_460916384.txt.gz]
|[https://drive.google.com/drive/folders/1HBPZ17llVE_8wCy5FvRUFQ5MJsaYXAW-?usp=drive_link Google Drive directory for 4x3 TMs]
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1310648046576730124 November 25, 2024 (@icy)]
|@tjligocki
|4,319
|[https://docs.google.com/spreadsheets/d/1grhW_0neb2I8TfceN5-v70_3W42Z2U159r9L6FhPGf8/edit?usp=sharing Spreadsheet of holdouts]
|Keeping track of BB(6) progress - informal.
|-
|[[BB(3,3)]]
|November 10, 2024
|
|6
|[[:File:3x3 holdout 6.txt|3x3 holdout 6.txt]]
|
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1304303803213942846 November 8, 2024]
|@mxdys
|4,408
|[[:File:BB6 holdouts 4408.txt|BB6_holdouts_4408.txt]]
|
|-
|[[BB(2,6)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1303219184221683733 November 4, 2024]
|@tjligocki
|22,302,296
|[https://drive.google.com/file/d/1xRLIjdiqImFP2SL38gvhxVlAaX0L1cYO/view?usp=drive_link 2x6_holdouts_22302296.txt.gz]
|[https://drive.google.com/drive/folders/1p9b5g-Id3WEMUYIwEnaKWRBGIW66ADjM?usp=drive_link Google Drive directory for 2x6 TMs]
|-
|[[BB(3,4)]]
|[https://discord.com/channels/960643023006490684/960643023530762341/1302767449476694188 November 3, 2024]
|@tjligocki
|434,787,751
|[https://drive.google.com/file/d/1PLzN3wLw-MRgk1OFmYh4RTwNc30nflR7/view?usp=drive_link 3x4_holdouts_434787751.txt.gz]
|[https://drive.google.com/drive/folders/1bZxl7jg5q9IVvHQNZwItx1kPusAWznZk?usp=drive_link Google Drive directory for 3x4 TMs]
|-
|[[BB(6)]]
|[https://discordapp.com/channels/960643023006490684/1239205785913790465/1280185195877634098 September 2, 2024]
|@mxdys
|5394
|[[:File:BB6 holdouts 5394.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1269612923127599164 August 4, 2024]
|@mxdys
|5877
|[[:File:BB6 holdouts 5877.txt]]
|
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1259131753176498216 July 6, 2024]
|@mxdys
|7296
|[[:File:BB6 holdouts 7296.txt]]
|
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1252989316175499284 June 19, 2024]
|@mxdys
|217
|[[:File:2x5_holdouts_217.txt]]
|273 holdouts minus machines solved by CTL
|-
|[[BB(2,5)]]
|June 15th 2024
|@dyuan01
| 273
|[[:File:2x5_holdouts_273.txt]]
|@Justin Blanchard's 499 holdouts minus machines solved by @mxdys
|-
|[[BB(6)]]
|[https://discord.com/channels/960643023006490684/1239205785913790465/1250895665719148595 June 13, 2024]
|@tjligocki
| 12,091
|[[:File:BB6 holdouts 12091.txt]]
|Work done with @Shawn Ligocki
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1249142547217907772 June 9, 2024]
|@Justin Blanchard
|22
|[[:File:3x3.todo.txt]], [[:File:Mugshots small.pdf]]
|
|-
|BB(6)
|[https://discord.com/channels/960643023006490684/1239205785913790465/1248708916381220954 June 7, 2024]
|@mxdys
|12,325
|[[:File:BB6 holdouts 12325.txt]]
|Some equivalent machines are removed.
|-
|[[BB(5)]]
|June 2024
|BBChallenge
|0
|
|'''BB(5) is SOLVED!'''
|-
|[[BB(2,5)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1242679236142170203 May 22, 2024]
|@Justin Blanchard
|499
|[[:File:2x5.todo.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|925
|[[:File:2023 06 08.3x3.holdouts intersect sligocki iijil 925.txt]]
|Intersection of @sligocki and @Iijil from below
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 June 8, 2023]
|@Iijil
|2,480
|[[:File:2023 06 08.3x3.holdouts iijil 2380.txt]]
|
|-
|[[BB(3,3)]]
|[https://discord.com/channels/960643023006490684/1084047886494470185/1116178334620070000 June 7, 2023]
|@sligocki
|2,417
|[[:File:2023 06 07.3x3.holdouts 2417.txt]]
|
|-
|[[BB(6)]]
|May 27, 2023
|@sligocki
|181,851
|[https://drive.google.com/file/d/1YNwFCN6XJeDNKxxK5KbGHOAFOdIBvDb6/view?usp=drive_link 6x2.holdouts_181851.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(6)]]
|May 10, 2023
|@sligocki
|1,458,704
|[https://drive.google.com/file/d/14bDnBt0OwuHATFBiubc_5jub0220EyXf/view?usp=drive_link 6x2.holdouts_1458704.txt.gz]
|This was posted to the BBChallenge Forum (before Discord)
|-
|[[BB(5)]]
|[https://skelet.ludost.net/bb/nreg.html circa May 13th, 2003]
|Georgi Georgiev (Skelet)
|43
|[https://bbchallenge.org/skelet List of 43 holdouts]
|
|}

TMBR: October 2025

2025-10-06T03:28:01Z

Qwertyasdf: Undo revision 4344 by Qwertyasdf (talk) nvm

{{TMBRnav|September 2025|November 2025}}''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).''
[[File:Wily Coyote Roadrunner Naming.png|thumb|[[Wily Coyote]], a [[BB(3,3)]] holdout]]

== Blog Posts ==

* ? Oct 2025. Katelyn Doucette. [https://katelyndoucette.com/articles/building-the-busy-beaver-ladder Building the Busy Beaver Ladder].

== Holdouts ==

* BB(7):
** Andrew Ducharme has continued reducing the [[BB(7)#Phase 2|number of holdouts]] with Stage 4 of Phase 2. Initially, in the beginning of the month there were 22,801,601 holdouts, and 22,721,168 holdouts remain. (0.35% reduction)
*BB(3,4):
**[[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]] are running filters in the domain under Phase 2, reducing the holdouts count from 434,787,751 to 64,777,377 (85.1% reduction).

== Theory ==
TODO: Update this section after studying existing literature a bit more.

[[Linear-Inequality Affine Transformation Automata]] (LIATA) were introduced as a generalization of the [[BMO1]] rules:

* @Bard proved that 3 dimension LIATA are Turing complete: [https://discord.com/channels/960643023006490684/1239205785913790465/1420457986564030641]
* @star proved that 2 dimension LIATA are Turing complete: [https://discord.com/channels/960643023006490684/1239205785913790465/1421271424588451915]
* BMO1 is a 2d-LIATA so this provides some sense for the difficulty of the problem.

[[Category:This Month in Beaver Research|2025-10]]

TMBR: October 2025

2025-10-06T03:27:28Z

Qwertyasdf: link

{{TMBRnav|September 2025|November 2025}}''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).''
[[File:Wily Coyote Roadrunner Naming.png|thumb|[[Wily Coyote]], a [[BB(3,3)]] holdout]]

== Blog Posts ==

* ? Oct 2025. Katelyn Doucette. [https://katelyndoucette.com/articles/building-the-busy-beaver-ladder Building the Busy Beaver Ladder].

== Holdouts ==

* BB(7):
** Andrew Ducharme has continued reducing the [[BB(7)#Phase 2|number of holdouts]] with Stage 4 of Phase 2. Initially, in the beginning of the month there were 22,801,601 holdouts, and 22,721,168 holdouts remain. (0.35% reduction)
*BB(3,4):
**[[User:XnoobSpeakable|XnoobSpeakable]] and [[User:WarpedWartWars|Lúkos]] are running filters in the domain under Phase 2, reducing the holdouts count from 434,787,751 to 64,777,377 (85.1% reduction).

== Theory ==
TODO: Update this section after studying existing literature a bit more.

[[Linear-Inequality Affine Transformation Automata]] (LIATA) were introduced as a generalization of the [[BMO1]] rules:

* @Bard proved that 3 dimension LIATA are Turing complete: [https://discord.com/channels/960643023006490684/1239205785913790465/1420457986564030641]
* @star proved that 2 dimension LIATA are Turing complete: [https://discord.com/channels/960643023006490684/1239205785913790465/1421271424588451915]
* [[BMO1]] is a 2d-LIATA so this provides some sense for the difficulty of the problem.

[[Category:This Month in Beaver Research|2025-10]]