BusyBeaverWiki - User contributions [en]

BB(2,5)

2025-10-25T19:34:39Z

Peacemaker II:

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

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

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

== Cryptids ==
Known Cryptids:

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

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

==Top Halters==
{{Incomplete List}}
Some of the longest running halting BB(2,5) TMs are:
{| class="wikitable"
|+
!Standard format
!(approximate) runtime
|-
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|<math>10 \uparrow\uparrow 4.8142742</math>
|-
|{{TM|1RB2LB4LB3LA1RZ_1LA3RA3LB0LB0RA|halt}}
|<math>> 10^{38\,033}</math>
|}

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

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

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

== Holdouts ==
This section is based on the list of 83 holdouts published by @mxdys, and includes further progress as of 25 October 2025.

=== Cryptids ===

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

=== Unsolved ===

* {{TM|1RB---4LB1RA4RA_2LB2LA3RA4LB0RB|undecided}}.
* {{TM|1RB---4LB0LA4RA_2LB2LA3RA4LB0RB|undecided}}.
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}. [https://discord.com/channels/960643023006490684/1354037062830919690/1354037062830919690 Shift overflow counter, potential Cryptid]
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}. Potential Cryptid
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}. Potential Cryptid
* {{TM|1RB3LA1LA4LA2RA_2LB2RA---0RA0RB|undecided}}.
* {{TM|1RB4RA1LA4RB2LA_2LB3LA1RB2RA---|undecided}}.
* {{TM|1RB---3RA2LA2RB_2LB3LA4LB4RA0RA|undecided}}.
* {{TM|1RB4LA1RA1RB1LA_2LB3LA---4RA2RB|undecided}}. [https://discord.com/channels/960643023006490684/1359561443929886760 Basic long. analysis by @dyuan]
* {{TM|1RB---4RB2RB4LA_2LB3LA3LB0RA0RB|undecided}}. [https://discord.com/channels/960643023006490684/1353983911222312970/1353983911222312970 Bouncer + chaotic counter]
* {{TM|1RB2RB3LA4LA1LA_2LB3RA---4RA1RB|undecided}}.
* {{TM|1RB3RB3LA4LA2RB_2LB3RA---1RA1LA|undecided}}.
* {{TM|1RB2LA4LA1RA1LA_2LB3RB4RB---2RA|undecided}}.
* {{TM|1RB4RB4RA1LA3LA_1LB2LA3RB2RB---|undecided}}.
* {{TM|1RB3RA3RB4LA1LA_1LB2LA1LA---1RB|undecided}}. [https://discord.com/channels/960643023006490684/1349602897024782358 Potentially halt]
* {{TM|1RB2RA4LA1RB4RB_1LB2LA3RA---0LB|undecided}}.
* {{TM|1RB3RB1LA2LA3RA_1LB2RA4RB0LA---|undecided}}.
* {{TM|1RB3LA1LA2RB2LB_1LB2RA4RA0RB---|undecided}}.
* {{TM|1RB2LA0RB0LB3LB_2LA4RB3RA0RA---|undecided}}.
* {{TM|1RB2RA3LB---2LB_2LA0LA4RB0RB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1329808777754706046 30% chance of beating current champion]
* {{TM|1RB2RA3LB4LA---_2LA0RB1LA2RB0RA|undecided}}. [https://discord.com/channels/960643023006490684/1348878717870673981 Analysis by @dyuan01 and @Legion]
* {{TM|1RB2RA3LA---2LB_2LA4RA4RB0RB0LA|undecided}}. Spaghetti [https://discord.com/channels/960643023006490684/1344221797020602398/1344221797020602398 analysis by @nerdyjoe]
* {{TM|1RB2RA3LA4LA2RB_2LA---3LB1RA3RA|undecided}}. [https://discord.com/channels/960643023006490684/1353983911222312970/1355112650690003028 Bouncer + chaotic counter]
* {{TM|1RB2RA3LA4LA2RB_2LA3RB---0RA1LA|undecided}}. Longitudinal analysis suggests chaotic
* {{TM|1RB3LA1RA4LA2RA_2LA---1LA0RA3RB|undecided}}. Longitudinal analysis suggests chaotic. [https://discord.com/channels/960643023006490684/1378560417235734558 More analysis by mxdys]
* {{TM|1RB3LA3LB0RB0LA_2LA4RB1LB1RA---|undecided}}.
* {{TM|1RB3LA1LA2RB2RA_2LA4RA3LB1RA---|undecided}}.
* {{TM|1RB3LA3LA0RB2LB_2LA4LA4RA2RA---|undecided}}. [https://discord.com/channels/960643023006490684/1376383949575557161 Analysis by @mxdys]
* {{TM|1RB2LB3LA0RA1LB_2LA4RA3RB3LA---|undecided}}. [https://discord.com/channels/960643023006490684/1344221797020602398/1344221797020602398 Analysis by @nerdyjoe]
* {{TM|1RB2RB---0LB3LA_2LA2LB3RB4RB1LB|undecided}}. Longitudinal analysis suggests chaotic
* {{TM|1RB2RB4LA2RA1LA_2LA4RA3LA---3RA|undecided}}. Longitudinal analysis suggests chaotic
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}}. Potential cryptid - [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]
* {{TM|1RB3LB0RB---2LB_2LA3RA4RB2RB0LA|undecided}}. [https://discord.com/channels/960643023006490684/1376577295938097253 Analysis by @mxdys]
* {{TM|1RB3LB4LA0LB---_2LA0LA1RB0RA3RA|undecided}}.
* {{TM|1RB3RB---4RA2RA_2LA2RA3LB4LB1LB|undecided}}.
* {{TM|1RB3RB1LB2RA---_2LA2RB1LA4LB0RA|undecided}}.
* {{TM|1RB2LA0RB1LA3LB_1LA3LB1RA4RA---|undecided}}.
* {{TM|1RB2LA0RB4LB1RA_1LA3RA1RA---0LA|undecided}}.
* {{TM|1RB2LA0RB---4LA_1LA3LA1RA4RA1LB|undecided}}.
* {{TM|1RB2LA0RB4LB0LA_1LA3LA1RA4RA---|undecided}}.
* {{TM|1RB2LA4RA1LA3LA_0LA2RB3RB2LB---|undecided}}. 1D CA-like. [https://discord.com/channels/960643023006490684/1354107790330691655 Analysis by @dyuan and @mxdys]
* {{TM|1RB2LA4RA1LA3LA_0LA3RB3LB2RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA1LA4RA2LA_0LA3RB3LB2RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA3LA4RA1LA_0LA3LB3RB1RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA3LB4LB---_0LA4LB3RA4LA0RB|undecided}}. Fractal?
* {{TM|1RB2RA3LA4LA2RB_2LA3RA---0RA1LA|undecided}}. Longitudinal analysis suggest chaotic
* {{TM|1RB---3LB4RB0LA_2LB3LA3RB4RA0RA|undecided}}.
The 15 grandchildren of {{TM|1RB2LA0RB1LB_1LA3RA1RA---|undecided}}

* {{TM|1RB2LA0RB1LB0LB_1LA3RA1RA4RA---|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4RB0LB|undecided}}.

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

* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2RB|undecided}}. Simulated for ~9e1167 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2LB|undecided}}. Simulated for ~1.3e1094 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1RB|undecided}}. Simulated for ~9.8e1226 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1LB|undecided}}. Simulated for ~3.0e1140 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0RB|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0LB|undecided}}. Simulated for ~2.6e889 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB3RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2LA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1LA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0LA|undecided}}.

=== Solved with moderate rigor ===

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

=== Formally proven ===

* {{TM|1RB3RA4LB2RA2RB_2LA---3LA0LB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 Rocq-decided by @mxdys.]
* {{TM|1RB3RB---0RA2RB_2LA4RA3LB1LB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 Rocq-decided by @mxdys.]
* {{TM|1RB2RA3LA4LA2RB_2LA0RA---0RA1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1355828077023854752 Rocq-decided by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1355714495326060706 Longitudinal analysis by @Legion implies nonhalting]
* {{TM|1RB0LB2LA4LB3LA_2LA---3RA4RB2RB|undecided}}. [https://discord.com/channels/960643023006490684/1375251026411786310/1375556785603084469 Rocq-decided by @mxdys]
* {{TM|1RB2LA1RA---1LA_1LA4RB3LB0RB2RB|undecided}}. [https://discord.com/channels/960643023006490684/1375251026411786310/1375556785603084469 Rocq-decided by @mxdys]
* {{TM|1RB2LA3LA4RA0LA_1LA3RB1RB1LB---|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379521528869421137 Rocq-decided by @mxdys]
* {{TM|1RB2RA3LA4RB---_2LA3RB3RA1LB3LB|halt}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 Rocq-decided by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1373347187836194898 Halting argument by @dyuan]
* {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 Rocq-decided by @mxdys]. Current champion

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

BB(2,5)

2025-10-25T19:33:28Z

Peacemaker II:

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

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

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

== Cryptids ==
Known Cryptids:

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

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

==Top Halters==
{{Incomplete List}}
Some of the longest running halting BB(2,5) TMs are:
{| class="wikitable"
|+
!Standard format
!(approximate) runtime
|-
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|<math>10 \uparrow\uparrow 4.8142742</math>
|-
|{{TM|1RB2LB4LB3LA1RZ_1LA3RA3LB0LB0RA|halt}}
|<math>> 10^{38\,033}</math>
|}

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

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

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

== Holdouts ==
This section is based on the list of 83 holdouts published by @mxdys, and includes further progress as of 25 October 2025.

=== Cryptids ===

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

=== Unsolved ===

* {{TM|1RB---4LB1RA4RA_2LB2LA3RA4LB0RB|undecided}}.
* {{TM|1RB---4LB0LA4RA_2LB2LA3RA4LB0RB|undecided}}.
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}. [https://discord.com/channels/960643023006490684/1354037062830919690/1354037062830919690 Shift overflow counter, potential Cryptid]
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}. Potential Cryptid
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}. Potential Cryptid
* {{TM|1RB3LA1LA4LA2RA_2LB2RA---0RA0RB|undecided}}.
* {{TM|1RB4RA1LA4RB2LA_2LB3LA1RB2RA---|undecided}}.
* {{TM|1RB---3RA2LA2RB_2LB3LA4LB4RA0RA|undecided}}.
* {{TM|1RB4LA1RA1RB1LA_2LB3LA---4RA2RB|undecided}}. [https://discord.com/channels/960643023006490684/1359561443929886760 Basic long. analysis by @dyuan]
* {{TM|1RB---4RB2RB4LA_2LB3LA3LB0RA0RB|undecided}}. [https://discord.com/channels/960643023006490684/1353983911222312970/1353983911222312970 Bouncer + chaotic counter]
* {{TM|1RB2RB3LA4LA1LA_2LB3RA---4RA1RB|undecided}}.
* {{TM|1RB3RB3LA4LA2RB_2LB3RA---1RA1LA|undecided}}.
* {{TM|1RB2LA4LA1RA1LA_2LB3RB4RB---2RA|undecided}}.
* {{TM|1RB4RB4RA1LA3LA_1LB2LA3RB2RB---|undecided}}.
* {{TM|1RB3RA3RB4LA1LA_1LB2LA1LA---1RB|undecided}}. [https://discord.com/channels/960643023006490684/1349602897024782358 Potentially halt]
* {{TM|1RB2RA4LA1RB4RB_1LB2LA3RA---0LB|undecided}}.
* {{TM|1RB3RB1LA2LA3RA_1LB2RA4RB0LA---|undecided}}.
* {{TM|1RB3LA1LA2RB2LB_1LB2RA4RA0RB---|undecided}}.
* {{TM|1RB2LA0RB0LB3LB_2LA4RB3RA0RA---|undecided}}.
* {{TM|1RB2RA3LB---2LB_2LA0LA4RB0RB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1329808777754706046 30% chance of beating current champion]
* {{TM|1RB2RA3LB4LA---_2LA0RB1LA2RB0RA|undecided}}. [https://discord.com/channels/960643023006490684/1348878717870673981 Analysis by @dyuan01 and @Legion]
* {{TM|1RB2RA3LA---2LB_2LA4RA4RB0RB0LA|undecided}}. Spaghetti [https://discord.com/channels/960643023006490684/1344221797020602398/1344221797020602398 analysis by @nerdyjoe]
* {{TM|1RB2RA3LA4LA2RB_2LA---3LB1RA3RA|undecided}}. [https://discord.com/channels/960643023006490684/1353983911222312970/1355112650690003028 Bouncer + chaotic counter]
* {{TM|1RB2RA3LA4LA2RB_2LA3RB---0RA1LA|undecided}}. Longitudinal analysis suggests chaotic
* {{TM|1RB3LA1RA4LA2RA_2LA---1LA0RA3RB|undecided}}. Longitudinal analysis suggests chaotic. [https://discord.com/channels/960643023006490684/1378560417235734558 More analysis by mxdys]
* {{TM|1RB3LA3LB0RB0LA_2LA4RB1LB1RA---|undecided}}.
* {{TM|1RB3LA1LA2RB2RA_2LA4RA3LB1RA---|undecided}}.
* {{TM|1RB3LA3LA0RB2LB_2LA4LA4RA2RA---|undecided}}. [https://discord.com/channels/960643023006490684/1376383949575557161 Analysis by @mxdys]
* {{TM|1RB2LB3LA0RA1LB_2LA4RA3RB3LA---|undecided}}. [https://discord.com/channels/960643023006490684/1344221797020602398/1344221797020602398 Analysis by @nerdyjoe]
* {{TM|1RB2RB---0LB3LA_2LA2LB3RB4RB1LB|undecided}}. Longitudinal analysis suggests chaotic
* {{TM|1RB2RB4LA2RA1LA_2LA4RA3LA---3RA|undecided}}. Longitudinal analysis suggests chaotic
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}}. Potential cryptid - [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]
* {{TM|1RB3LB0RB---2LB_2LA3RA4RB2RB0LA|undecided}}. [https://discord.com/channels/960643023006490684/1376577295938097253 Analysis by @mxdys]
* {{TM|1RB3LB4LA0LB---_2LA0LA1RB0RA3RA|undecided}}.
* {{TM|1RB3RB---4RA2RA_2LA2RA3LB4LB1LB|undecided}}.
* {{TM|1RB3RB1LB2RA---_2LA2RB1LA4LB0RA|undecided}}.
* {{TM|1RB2LA0RB1LA3LB_1LA3LB1RA4RA---|undecided}}.
* {{TM|1RB2LA0RB4LB1RA_1LA3RA1RA---0LA|undecided}}.
* {{TM|1RB2LA0RB---4LA_1LA3LA1RA4RA1LB|undecided}}.
* {{TM|1RB2LA0RB4LB0LA_1LA3LA1RA4RA---|undecided}}.
* {{TM|1RB2LA4RA1LA3LA_0LA2RB3RB2LB---|undecided}}. 1D CA-like. [https://discord.com/channels/960643023006490684/1354107790330691655 Analysis by @dyuan and @mxdys]
* {{TM|1RB2LA4RA1LA3LA_0LA3RB3LB2RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA1LA4RA2LA_0LA3RB3LB2RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA3LA4RA1LA_0LA3LB3RB1RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA3LB4LB---_0LA4LB3RA4LA0RB|undecided}}. Fractal?
* {{TM|1RB2RA3LA4LA2RB_2LA3RA---0RA1LA|undecided}}. Longitudinal analysis suggest chaotic
* {{TM|1RB---3LB4RB0LA_2LB3LA3RB4RA0RA|undecided}}.
The 15 grandchildren of {{TM|1RB2LA0RB1LB_1LA3RA1RA---|undecided}}

* {{TM|1RB2LA0RB1LB0LB_1LA3RA1RA4RA---|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4RB0LB|undecided}}.

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

* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2RB|undecided}}. Simulated for ~9e1167 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2LB|undecided}}. Simulated for ~1.3e1094 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1RB|undecided}}. Simulated for ~9.8e1226 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1LB|undecided}}. Simulated for ~3.0e1140 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0RB|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0LB|undecided}}. Simulated for ~2.6e889 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB3RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2LA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1LA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0LA|undecided}}.

=== Solved with moderate rigor ===

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

=== Formally proven ===

* {{TM|1RB3RA4LB2RA2RB_2LA---3LA0LB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 Rocq-decided by @mxdys.]
* {{TM|1RB3RB---0RA2RB_2LA4RA3LB1LB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 Rocq-decided by @mxdys.]
* {{TM|1RB2RA3LA4LA2RB_2LA0RA---0RA1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1355828077023854752 Rocq-decided by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1355714495326060706 Longitudinal analysis by @Legion implies nonhalting]
* {{TM|1RB0LB2LA4LB3LA_2LA---3RA4RB2RB|undecided}}. [https://discord.com/channels/960643023006490684/1375251026411786310/1375556785603084469 Rocq-decided by @mxdys]
* {{TM|1RB2LA1RA---1LA_1LA4RB3LB0RB2RB|undecided}}. [https://discord.com/channels/960643023006490684/1375251026411786310/1375556785603084469 Rocq-decided by @mxdys]
* {{TM|1RB2LA3LA4RA0LA_1LA3RB1RB1LB---|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379521528869421137 Rocq-decided by @mxdys]
* {{TM|1RB2RA3LA4RB---_2LA3RB3RA1LB3LB|halt}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 Rocq-decided by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1373347187836194898 Halting argument by @dyuan]
* {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 Rocq-decided by @mxdys]. Current champion

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

BB(2,5)

2025-10-25T19:29:59Z

Peacemaker II: minor changes

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

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

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

== Cryptids ==
Known Cryptids:

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

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

==Top Halters==
{{Incomplete List}}
Some of the longest running halting BB(2,5) TMs are:
{| class="wikitable"
|+
!Standard format
!(approximate) runtime
|-
|{{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}
|<math>10 \uparrow\uparrow 4.8142742</math>
|-
|{{TM|1RB2LB4LB3LA1RZ_1LA3RA3LB0LB0RA|halt}}
|<math>> 10^{38\,033}</math>
|}

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

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

Over the course of 5 months, @mxdys added 8 machines to Rocq[https://discord.com/channels/960643023006490684/1259770421046411285/1355799763437752521 1][https://discord.com/channels/960643023006490684/1259770421046411285/1355828077023854752 2][https://discord.com/channels/960643023006490684/1259770421046411285/1379521528869421137 3][https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 4][https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 5], lowering the certified holdout count to 75. There are 11 informal arguments (9 by equivalence classes), lowering the informal holdout count to 66 by equivalence classes.

== Holdouts ==
This section is based on progress as of 25 October 2025.

=== Cryptids ===

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

=== Unsolved ===

* {{TM|1RB---4LB1RA4RA_2LB2LA3RA4LB0RB|undecided}}.
* {{TM|1RB---4LB0LA4RA_2LB2LA3RA4LB0RB|undecided}}.
* {{TM|1RB---0RB0LA2RA_2LB2LA3RA4LB0LB|undecided}}. [https://discord.com/channels/960643023006490684/1354037062830919690/1354037062830919690 Shift overflow counter, potential Cryptid]
* {{TM|1RB3LA1LA1RA3RA_2LB2RA---4RB1LB|undecided}}. Potential Cryptid
* {{TM|1RB3LA1LA1RA1RA_2LB2RA---4RB1LB|undecided}}. Potential Cryptid
* {{TM|1RB3LA1LA4LA2RA_2LB2RA---0RA0RB|undecided}}.
* {{TM|1RB4RA1LA4RB2LA_2LB3LA1RB2RA---|undecided}}.
* {{TM|1RB---3RA2LA2RB_2LB3LA4LB4RA0RA|undecided}}.
* {{TM|1RB4LA1RA1RB1LA_2LB3LA---4RA2RB|undecided}}. [https://discord.com/channels/960643023006490684/1359561443929886760 Basic long. analysis by @dyuan]
* {{TM|1RB---4RB2RB4LA_2LB3LA3LB0RA0RB|undecided}}. [https://discord.com/channels/960643023006490684/1353983911222312970/1353983911222312970 Bouncer + chaotic counter]
* {{TM|1RB2RB3LA4LA1LA_2LB3RA---4RA1RB|undecided}}.
* {{TM|1RB3RB3LA4LA2RB_2LB3RA---1RA1LA|undecided}}.
* {{TM|1RB2LA4LA1RA1LA_2LB3RB4RB---2RA|undecided}}.
* {{TM|1RB4RB4RA1LA3LA_1LB2LA3RB2RB---|undecided}}.
* {{TM|1RB3RA3RB4LA1LA_1LB2LA1LA---1RB|undecided}}. [https://discord.com/channels/960643023006490684/1349602897024782358 Potentially halt]
* {{TM|1RB2RA4LA1RB4RB_1LB2LA3RA---0LB|undecided}}.
* {{TM|1RB3RB1LA2LA3RA_1LB2RA4RB0LA---|undecided}}.
* {{TM|1RB3LA1LA2RB2LB_1LB2RA4RA0RB---|undecided}}.
* {{TM|1RB2LA0RB0LB3LB_2LA4RB3RA0RA---|undecided}}.
* {{TM|1RB2RA3LB---2LB_2LA0LA4RB0RB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1329808777754706046 30% chance of beating current champion]
* {{TM|1RB2RA3LB4LA---_2LA0RB1LA2RB0RA|undecided}}. [https://discord.com/channels/960643023006490684/1348878717870673981 Analysis by @dyuan01 and @Legion]
* {{TM|1RB2RA3LA---2LB_2LA4RA4RB0RB0LA|undecided}}. Spaghetti [https://discord.com/channels/960643023006490684/1344221797020602398/1344221797020602398 analysis by @nerdyjoe]
* {{TM|1RB2RA3LA4LA2RB_2LA---3LB1RA3RA|undecided}}. [https://discord.com/channels/960643023006490684/1353983911222312970/1355112650690003028 Bouncer + chaotic counter]
* {{TM|1RB2RA3LA4LA2RB_2LA3RB---0RA1LA|undecided}}. Longitudinal analysis suggests chaotic
* {{TM|1RB3LA1RA4LA2RA_2LA---1LA0RA3RB|undecided}}. Longitudinal analysis suggests chaotic. [https://discord.com/channels/960643023006490684/1378560417235734558 More analysis by mxdys]
* {{TM|1RB3LA3LB0RB0LA_2LA4RB1LB1RA---|undecided}}.
* {{TM|1RB3LA1LA2RB2RA_2LA4RA3LB1RA---|undecided}}.
* {{TM|1RB3LA3LA0RB2LB_2LA4LA4RA2RA---|undecided}}. [https://discord.com/channels/960643023006490684/1376383949575557161 Analysis by @mxdys]
* {{TM|1RB2LB3LA0RA1LB_2LA4RA3RB3LA---|undecided}}. [https://discord.com/channels/960643023006490684/1344221797020602398/1344221797020602398 Analysis by @nerdyjoe]
* {{TM|1RB2RB---0LB3LA_2LA2LB3RB4RB1LB|undecided}}. Longitudinal analysis suggests chaotic
* {{TM|1RB2RB4LA2RA1LA_2LA4RA3LA---3RA|undecided}}. Longitudinal analysis suggests chaotic
* {{TM|1RB3LB---4LA1RB_2LA4LA4LB3RB1RA|undecided}}. Potential cryptid - [https://discord.com/channels/960643023006490684/1375584513777995957 Analysis by @mxdys]
* {{TM|1RB3LB0RB---2LB_2LA3RA4RB2RB0LA|undecided}}. [https://discord.com/channels/960643023006490684/1376577295938097253 Analysis by @mxdys]
* {{TM|1RB3LB4LA0LB---_2LA0LA1RB0RA3RA|undecided}}.
* {{TM|1RB3RB---4RA2RA_2LA2RA3LB4LB1LB|undecided}}.
* {{TM|1RB3RB1LB2RA---_2LA2RB1LA4LB0RA|undecided}}.
* {{TM|1RB2LA0RB1LA3LB_1LA3LB1RA4RA---|undecided}}.
* {{TM|1RB2LA0RB4LB1RA_1LA3RA1RA---0LA|undecided}}.
* {{TM|1RB2LA0RB---4LA_1LA3LA1RA4RA1LB|undecided}}.
* {{TM|1RB2LA0RB4LB0LA_1LA3LA1RA4RA---|undecided}}.
* {{TM|1RB2LA4RA1LA3LA_0LA2RB3RB2LB---|undecided}}. 1D CA-like. [https://discord.com/channels/960643023006490684/1354107790330691655 Analysis by @dyuan and @mxdys]
* {{TM|1RB2LA4RA1LA3LA_0LA3RB3LB2RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA1LA4RA2LA_0LA3RB3LB2RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA3LA4RA1LA_0LA3LB3RB1RB---|undecided}}. 1D CA-like
* {{TM|1RB2LA3LB4LB---_0LA4LB3RA4LA0RB|undecided}}. Fractal?
* {{TM|1RB2RA3LA4LA2RB_2LA3RA---0RA1LA|undecided}}. Longitudinal analysis suggest chaotic
* {{TM|1RB---3LB4RB0LA_2LB3LA3RB4RA0RA|undecided}}.
The 15 grandchildren of {{TM|1RB2LA0RB1LB_1LA3RA1RA---|undecided}}

* {{TM|1RB2LA0RB1LB0LB_1LA3RA1RA4RA---|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4RB0LB|undecided}}.

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

* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2RB|undecided}}. Simulated for ~9e1167 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2LB|undecided}}. Simulated for ~1.3e1094 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1RB|undecided}}. Simulated for ~9.8e1226 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1LB|undecided}}. Simulated for ~3.0e1140 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0RB|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0LB|undecided}}. Simulated for ~2.6e889 steps by @hipparcos, [https://discord.com/channels/960643023006490684/1336734852308799579/1352407027804143726 hasn't halted yet]
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB3RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB2LA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB1LA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0RA|undecided}}.
* {{TM|1RB2LA0RB1LB---_1LA3RA1RA4LB0LA|undecided}}.

=== Solved with moderate rigor ===

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

=== Formally proven ===

* {{TM|1RB3RA4LB2RA2RB_2LA---3LA0LB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 Rocq-decided by @mxdys.]
* {{TM|1RB3RB---0RA2RB_2LA4RA3LB1LB1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1411488532500971631 Rocq-decided by @mxdys.]
* {{TM|1RB2RA3LA4LA2RB_2LA0RA---0RA1LA|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1355828077023854752 Rocq-decided by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1355714495326060706 Longitudinal analysis by @Legion implies nonhalting]
* {{TM|1RB0LB2LA4LB3LA_2LA---3RA4RB2RB|undecided}}. [https://discord.com/channels/960643023006490684/1375251026411786310/1375556785603084469 Rocq-decided by @mxdys]
* {{TM|1RB2LA1RA---1LA_1LA4RB3LB0RB2RB|undecided}}. [https://discord.com/channels/960643023006490684/1375251026411786310/1375556785603084469 Rocq-decided by @mxdys]
* {{TM|1RB2LA3LA4RA0LA_1LA3RB1RB1LB---|undecided}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379521528869421137 Rocq-decided by @mxdys]
* {{TM|1RB2RA3LA4RB---_2LA3RB3RA1LB3LB|halt}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 Rocq-decided by @mxdys]. [https://discord.com/channels/960643023006490684/1259770421046411285/1373347187836194898 Halting argument by @dyuan]
* {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA---|halt}}. [https://discord.com/channels/960643023006490684/1259770421046411285/1379877629288644722 Rocq-decided by @mxdys]. Current champion

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

BB(3,3)

2025-10-13T00:02:28Z

Peacemaker II: /* Certified Progress */

The 3-state, 3-symbol Busy Beaver problem, '''BB(3,3)''', is unsolved. With the discovery of the [[Cryptids|Cryptid]] machine [[Bigfoot]] in October 2023, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(3,3) and thus [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html BB(3,3) is Hard]. 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>

One of the currently unsolved TMs, {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}}, is under exploration on Discord and believed to [[probviously]] halt. If it halts, it will be the new champion.

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

* {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}}

== 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 August 2025 [[Holdouts lists|holdouts list]] of [https://discord.com/channels/960643023006490684/1259770474897080380/1410308974275985428 10-1 TMs.] [https://discord.com/channels/960643023006490684/1259770474897080380/1422714899183964240 (-1)]

=== 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 ===
* {{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]

=== 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].

== 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 four 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|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)
* {{TM|1RB2RA1LB_0LC0RA1LA_2LA0RB---|undecided}} (818, equivalent to 21 and 92)
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)

2025-10-13T00:00:47Z

Peacemaker II: /* Certified Progress */

The 3-state, 3-symbol Busy Beaver problem, '''BB(3,3)''', is unsolved. With the discovery of the [[Cryptids|Cryptid]] machine [[Bigfoot]] in October 2023, we now know that we must solve a [[Collatz-like]] problem in order to solve BB(3,3) and thus [https://www.sligocki.com/2023/10/16/bb-3-3-is-hard.html BB(3,3) is Hard]. 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>

One of the currently unsolved TMs, {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}}, is under exploration on Discord and believed to [[probviously]] halt. If it halts, it will be the new champion.

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

* {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}}

== 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 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 August 2025 [[Holdouts lists|holdouts list]] of [https://discord.com/channels/960643023006490684/1259770474897080380/1410308974275985428 10-1 TMs.] [https://discord.com/channels/960643023006490684/1259770474897080380/1422714899183964240 (-1)]

=== 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 ===
* {{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]

=== 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].

== 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 four 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|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)
* {{TM|1RB2RA1LB_0LC0RA1LA_2LA0RB---|undecided}} (818, equivalent to 21 and 92)
These TMs were on Justin Blanchard's informal [[Holdouts lists|holdouts list]] of 22 TMs but were Rocq-decided individually [https://discord.com/channels/960643023006490684/1259770474897080380/1344554618188730368 by @mxdys] in their February 2025 release. Two other members of Justin Blanchard's list Rocq-decided by mxdys in February 2025 were {{TM|1RB2LB---_1RC2RB1LC_0LA0RB1LB|undecided}} (642) and {{TM|1RB2RB---_1LC2LB1RC_0RA0LB1RB|undecided}} (834). @-d independently generated a Rocq proof for 642 ([https://cosearch.bbchallenge.org/contribution/3wxkcx6f cosearch]), and @dyuan01 independently discovered non-halting arguments for [https://discord.com/channels/960643023006490684/1084047886494470185/1222694817771814922 642] and [https://discord.com/channels/960643023006490684/1084047886494470185/1222694817771814922 834], and noted their similarity.

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

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

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

User:Peacemaker II

2025-10-11T16:24:06Z

Peacemaker II:

discord: @peacemaker2

User:Peacemaker II

2025-10-11T16:23:28Z

Peacemaker II: Created page with "discord username: peacemaker2"

discord username: peacemaker2

BB(3,3)

2025-10-10T16:05:50Z

Peacemaker II: moved 867

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>

One of the currently unsolved TMs, {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}}, is under exploration on Discord and believed to [[probviously]] halt. If it halts, it will be the new champion.

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

* {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}}

== 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 [https://discord.com/channels/960643023006490684/1259770474897080380/1325592156257124443 5 Jan 2025], @tjligocki finished an enumeration and filtering of the BB(3,3) machines using the established Ligocki filters, listing the filter used for each machine. He also computed the number of steps and sigma scores for all found halting TMs. The thorough results are located [https://drive.google.com/drive/folders/101U9htddFdE889agAZnU58VLgQJlfn0v?usp=drive_link here]. 367 machines remained on that list. These results were updated on [https://discord.com/channels/960643023006490684/1259770474897080380/1352398469456859167 20 Mar 2025] in the same location and 76 machines remained on the list.

Over two-thirds of the 367 remaining machines were shown to be non-halting with FAR and MITMWFAR by @Justin Blanchard on [https://discord.com/channels/960643023006490684/1259770474897080380/1262265485639286876 14 July 2024]. Most of the remainders were shown non-halting by @lijil on [https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 8 June 2023.] Together, this leaves 21 unsolved TMs, all of which were on @Justin Blanchard's informal holdouts list of 22 machines. The extra machine on Justin's list {{TM|1RB2LA0LA_2LC---2RA_0RA2RC1LC|undecided}} had been solved by @lijil before, but this was not realized for some time.

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. All 4 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 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 ===
* {{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]

=== Formally proven ===
On August 30, 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].

== 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 four 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|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)
* {{TM|1RB2RA1LB_0LC0RA1LA_2LA0RB---|undecided}} (818, equivalent to 21 and 92)
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)]]

Talk:1RB1RD1LC 2LB1RB1LC 1RZ1LA1LD 2RB2RA2RD

2025-10-05T17:12:25Z

Peacemaker II: Created page with "== Score == The final TM score should be 10^^9.873987, not 10^^10.873987 ~~~~"

== Score ==
The final TM score should be 10^^9.873987, not 10^^10.873987 [[User:Peacemaker II|Peacemaker II]] ([[User talk:Peacemaker II|talk]]) 17:12, 5 October 2025 (UTC)

User talk:Polygon/Page for testing

2025-10-05T15:29:12Z

Peacemaker II: /* Comment */

== Comment ==
I know this page is very much a WIP but I think you meant "1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD" as the new BB(4,3) champion. Nice work on the analysis by the way! [[User:Peacemaker II|Peacemaker II]] ([[User talk:Peacemaker II|talk]]) 15:27, 5 October 2025 (UTC)
: Never mind, looks like you fixed it! [[User:Peacemaker II|Peacemaker II]] ([[User talk:Peacemaker II|talk]]) 15:29, 5 October 2025 (UTC)

User talk:Polygon/Page for testing

2025-10-05T15:27:36Z

Peacemaker II: Created page with "== Comment == I know this page is very much a WIP but I think you meant "1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD" as the new BB(4,3) champion. Nice work on the analysis by the way! ~~~~"

BB(2,6)

2025-09-28T10:22:39Z

Peacemaker II: /* Top Halters */ fix formatting

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

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

== Top Halters ==
{{Incomplete List}}
The scores are given using [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]] with an extension to decimal tetration<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>. The highest known scoring machines are:
{| class="wikitable"
|+
!TM
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|10 ↑↑↑ 3
|Pavel Kropitz
|-
|{{TM|1RB3LA4LB0RB1RA3LA_2LA2RA4LA1RA5RB1RZ|halt}}
|10 ↑↑ 91.17
|Pavel Kropitz
|-
|{{TM|1RB2LA1RA4LA5RA0LB_1LA3RA2RB1RZ3RB4LA|halt}}
|10 ↑↑ 70.27
|Shawn Ligocki
|-
|{{TM|1RB2LB1RZ3LA2LA4RB_1LA3RB4RB1LB5LB0RA|halt}}
|10 ↑↑ 69.68
|Shawn Ligocki
|-
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|Andrew Ducharme
|-
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|Andrew Ducharme
|-
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|Andrew Ducharme
|-
|{{TM|1RB3LB3RB4LA2LA4LA_2LA2RB1LB0RA5RA1RZ|halt}}
|10 ↑↑ 21.54
|Shawn Ligocki
|-
|{{TM|1RB2LB3LA1RA0RA1RZ_1LA2RB1LB4RB5RA3LA|halt}}
|10 ↑↑ 20.58
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB1RA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB5LA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Andrew Ducharme
|-
|{{TM|1RB3RA4LB5RA5LB4RA_2LA1RZ1RB2LA5LA0LA|halt}}
|10 ↑↑ 17.08
|Andrew Ducharme
|-
|{{TM|1RB3RA4LA1LA0LA1RZ_2LA0LB1RA1LB5LB2RA|halt}}
|10 ↑↑ 15.44
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ2LA|halt}}
|10 ↑↑ 14.35
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ3RA|halt}}
|10 ↑↑ 14.17
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ1LA|halt}}
|10 ↑↑ 14.05
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ0RA|halt}}
|10 ↑↑ 13.69
|Andrew Ducharme
|-
|{{TM|1RB3LA3RA4LB2LB0LA_2LA5LB2RB0RA0RA1RZ|halt}}
|10 ↑↑ 12.42
|Andrew Ducharme
|-
|{{TM|1RB0LB4LA2RA2RB1LB_2LA4LA3LB5LA1RA1RZ|halt}}
|10 ↑↑ 11.70
|Andrew Ducharme
|-
|{{TM|1RB2LA1RA1RA5LB1RZ_1LA4RB3RB4LB0RB0RA|halt}}
|10 ↑↑ 10.39
|Andrew Ducharme
|}
All decimal places are truncated.

== Filtering ==
Starting from Terry Ligocki's [[holdouts list]] of 22,302,296 TMs, additional filtering has been performed:

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

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|22,302,296
|20,246,662
|9.2%
|23.0
|24.80
|269.09
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 6 run
|rowspan="100" |[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=share_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|20,246,662
|19,134,631
|5.5%
|83.4
|3.71
|67.46
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|19,134,631
|5,443,318
|71.6%
|46.6
|81.69
|114.17
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|5,443,318
|3,400,118
|37.5%
|16.3
|34.87
|92.90
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|3,400,118
|3,303,416
|2.8%
|14.1
|1.90
|66.82
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 8 mod 3 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|3,303,416
|3,249,427
|1.6%
|16.5
|0.91
|55.59
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n 7 run
|-
|style="text-align:left" |Terry Ligocki
|3,249,427
|3,155,741
|2.9%
|13.4
|1.94
|67.25
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|3,155,741
|2,246,891
|28.8%
|14.1
|17.93
|62.24
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|2,246,891
|2,143,803
|4.6%
|7.5
|3.82
|83.19
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|2,143,803
|1,938,663
|9.6%
|7.5
|7.58
|79.21
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,938,663
|1,885,153
|2.8%
|7.5
|1.98
|71.72
|style="text-align:left" |chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,885,153
|1,848,887
|1.9%
|10.5
|0.96
|49.68
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|1,848,887
|1,816,027
|1.8%
|7.6
|1.20
|67.77
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 10000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,816,027
|1,688,951
|7.0%
|10.4
|3.40
|48.66
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 2 tH 0 n 2 run
|}
==References==


[[Category: BB Domains]]

BB(2,6)

2025-09-28T10:21:03Z

Peacemaker II: /* Top Halters */ fix formatting

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

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

== Top Halters ==
{{Incomplete List}}
The scores are given using [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]] with an extension to decimal tetration<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>. The highest known scoring machines are:
{| class="wikitable"
|+
!TM
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|10 ↑↑↑ 3
|Pavel Kropitz
|-
|{{TM|1RB3LA4LB0RB1RA3LA_2LA2RA4LA1RA5RB1RZ|halt}}
|10 ↑↑ 91.17
|Pavel Kropitz
|-
|{{TM|1RB2LA1RA4LA5RA0LB_1LA3RA2RB1RZ3RB4LA|halt}}
|10 ↑↑ 70.27
|Shawn Ligocki
|-
|{{TM|1RB2LB1RZ3LA2LA4RB_1LA3RB4RB1LB5LB0RA|halt}}
|10 ↑↑ 69.68
|Shawn Ligocki
|-
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|Andrew Ducharme
|-
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|Andrew Ducharme
|-
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|Andrew Ducharme
|-
|{{TM|1RB3LB3RB4LA2LA4LA_2LA2RB1LB0RA5RA1RZ|halt}}
|10 ↑↑ 21.54
|Shawn Ligocki
|-
|{{TM|1RB2LB3LA1RA0RA1RZ_1LA2RB1LB4RB5RA3LA|halt}}
|10 ↑↑ 20.58
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB1RA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB5LA2LA_2LA4LB1RA3LB5LB1RZ}|halt}}
|10 ↑↑ 17.53
|Andrew Ducharme
|-
|{{TM|1RB3RA4LB5RA5LB4RA_2LA1RZ1RB2LA5LA0LA}|halt}}
|10 ↑↑ 17.08
|Andrew Ducharme
|-
|{{TM|1RB3RA4LA1LA0LA1RZ_2LA0LB1RA1LB5LB2RA}|halt}}
|10 ↑↑ 15.44
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ2LA}|halt}}
|10 ↑↑ 14.35
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ3RA}|halt}}
|10 ↑↑ 14.17
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ1LA}|halt}}
|10 ↑↑ 14.05
|Andrew Ducharme
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ0RA}|halt}}
|10 ↑↑ 13.69
|Andrew Ducharme
|-
|{{TM|1RB3LA3RA4LB2LB0LA_2LA5LB2RB0RA0RA1RZ}|halt}}
|10 ↑↑ 12.42
|Andrew Ducharme
|-
|{{TM|1RB0LB4LA2RA2RB1LB_2LA4LA3LB5LA1RA1RZ}|halt}}
|10 ↑↑ 11.70
|Andrew Ducharme
|-
|{{TM|1RB2LA1RA1RA5LB1RZ_1LA4RB3RB4LB0RB0RA}|halt}}
|10 ↑↑ 10.39
|Andrew Ducharme
|}
All decimal places are truncated.

== Filtering ==
Starting from Terry Ligocki's [[holdouts list]] of 22,302,296 TMs, additional filtering has been performed:

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

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|22,302,296
|20,246,662
|9.2%
|23.0
|24.80
|269.09
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 6 run
|rowspan="100" |[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=share_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|20,246,662
|19,134,631
|5.5%
|83.4
|3.71
|67.46
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|19,134,631
|5,443,318
|71.6%
|46.6
|81.69
|114.17
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|5,443,318
|3,400,118
|37.5%
|16.3
|34.87
|92.90
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|3,400,118
|3,303,416
|2.8%
|14.1
|1.90
|66.82
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 8 mod 3 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|3,303,416
|3,249,427
|1.6%
|16.5
|0.91
|55.59
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n 7 run
|-
|style="text-align:left" |Terry Ligocki
|3,249,427
|3,155,741
|2.9%
|13.4
|1.94
|67.25
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|3,155,741
|2,246,891
|28.8%
|14.1
|17.93
|62.24
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|2,246,891
|2,143,803
|4.6%
|7.5
|3.82
|83.19
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|2,143,803
|1,938,663
|9.6%
|7.5
|7.58
|79.21
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,938,663
|1,885,153
|2.8%
|7.5
|1.98
|71.72
|style="text-align:left" |chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,885,153
|1,848,887
|1.9%
|10.5
|0.96
|49.68
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|1,848,887
|1,816,027
|1.8%
|7.6
|1.20
|67.77
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 10000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,816,027
|1,688,951
|7.0%
|10.4
|3.40
|48.66
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 2 tH 0 n 2 run
|}
==References==


[[Category: BB Domains]]

BB(2,6)

2025-09-28T10:18:36Z

Peacemaker II: /* Top Halters */ fill list

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

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

== Top Halters ==
{{Incomplete List}}
The scores are given using [[wikipedia:Knuth's_up-arrow_notation|Knuth's up-arrow notation]] with an extension to decimal tetration<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>. The highest known scoring machines are:
{| class="wikitable"
|+
!TM
!Approximate sigma score
!Discoverer
|-
|{{TM|1RB3RB5RA1LB5LA2LB_2LA2RA4RB1RZ3LB2LA|halt}}
|10 ↑↑↑ 3
|Pavel Kropitz
|-
|{{TM|1RB3LA4LB0RB1RA3LA_2LA2RA4LA1RA5RB1RZ|halt}}
|10 ↑↑ 91.17
|Pavel Kropitz
|-
|{{TM|1RB2LA1RA4LA5RA0LB_1LA3RA2RB1RZ3RB4LA|halt}}
|10 ↑↑ 70.27
|Shawn Ligocki
|-
|{{TM|1RB2LB1RZ3LA2LA4RB_1LA3RB4RB1LB5LB0RA|halt}}
|10 ↑↑ 69.68
|Shawn Ligocki
|-
|{{TM|1RB2LB0RA2RA5RA1LB_2LA4RB3LB2RB0RB1RZ|halt}}
|10 ↑↑ 54.90
|Andrew Ducharme
|-
|{{TM|1RB3RB1LB5LA2LB1RZ_2LA3RA4RB2LB0LA4RB|halt}}
|10 ↑↑ 42.17
|Andrew Ducharme
|-
|{{TM|1RB3LB0RB5RA1LB1RZ_2LB3LA4RA0RB0RA2LB|halt}}
|10 ↑↑ 40.07
|Andrew Ducharme
|-
|{{TM|1RB3LB3RB4LA2LA4LA_2LA2RB1LB0RA5RA1RZ|halt}}
|10 ↑↑ 21.54
|Shawn Ligocki
|-
|{{TM|1RB2LB3LA1RA0RA1RZ_1LA2RB1LB4RB5RA3LA|halt}}
|10 ↑↑ 20.58
|Shawn Ligocki
|-
|{{TM|1RB0RA3RB0LB1RA2LA_2LA4LB1RA3LB5LB1RZ|halt}}
|10 ↑↑ 17.53
|Shawn Ligocki
|}
|-
|{{TM|1RB0RA3RB0LB5LA2LA_2LA4LB1RA3LB5LB1RZ}|halt}}
|10 ↑↑ 17.53
|Andrew Ducharme
|}
|-
|{{TM|1RB3RA4LB5RA5LB4RA_2LA1RZ1RB2LA5LA0LA}|halt}}
|10 ↑↑ 17.08
|Andrew Ducharme
|}
|-
|{{TM|1RB3RA4LA1LA0LA1RZ_2LA0LB1RA1LB5LB2RA}|halt}}
|10 ↑↑ 15.44
|Andrew Ducharme
|}
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ2LA}|halt}}
|10 ↑↑ 14.35
|Andrew Ducharme
|}
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ3RA}|halt}}
|10 ↑↑ 14.17
|Andrew Ducharme
|}
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ1LA}|halt}}
|10 ↑↑ 14.05
|Andrew Ducharme
|}
|-
|{{TM|1RB3RB5LA1LA2RA3LA_2LA3RA2LB4LB1RZ0RA}|halt}}
|10 ↑↑ 13.69
|Andrew Ducharme
|}
|-
|{{TM|1RB3LA3RA4LB2LB0LA_2LA5LB2RB0RA0RA1RZ}|halt}}
|10 ↑↑ 12.42
|Andrew Ducharme
|}
|-
|{{TM|1RB0LB4LA2RA2RB1LB_2LA4LA3LB5LA1RA1RZ}|halt}}
|10 ↑↑ 11.70
|Andrew Ducharme
|}
|-
|{{TM|1RB2LA1RA1RA5LB1RZ_1LA4RB3RB4LB0RB0RA}|halt}}
|10 ↑↑ 10.39
|Andrew Ducharme
|}
All decimal places are truncated.

== Filtering ==
Starting from Terry Ligocki's [[holdouts list]] of 22,302,296 TMs, additional filtering has been performed:

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

{| class="wikitable sortable" style="text-align: right"
!rowspan="2" |Done by
!colspan="2" |Holdout TMs
!rowspan="2" |<math>*^1</math>
!rowspan="2" |<math>*^2</math>
!colspan="2" |TMs/sec/core
!rowspan="2" |Description
!rowspan="2" |Data
|-
!Input
!Output
!<math>*^3</math>
!<math>*^4</math>
|-
|style="text-align:left" |Terry Ligocki
|22,302,296
|20,246,662
|9.2%
|23.0
|24.80
|269.09
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 3000 H 6 mod 2 n 6 run
|rowspan="100" |[https://drive.google.com/drive/folders/1TsSpW27x3LBlu5qmk-cjzCJzgo_3ehyT?usp=share_link Google Drive]
|-
|style="text-align:left" |Terry Ligocki
|20,246,662
|19,134,631
|5.5%
|83.4
|3.71
|67.46
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 8 run
|-
|style="text-align:left" |Terry Ligocki
|19,134,631
|5,443,318
|71.6%
|46.6
|81.69
|114.17
|style="text-align:left" |chr_LRUH 20 chr_H 12 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|5,443,318
|3,400,118
|37.5%
|16.3
|34.87
|92.90
|style="text-align:left" |chr_LRUH 8 chr_H 4 MitM_CTL NG maxT 10000 NG_n 3 run
|-
|style="text-align:left" |Terry Ligocki
|3,400,118
|3,303,416
|2.8%
|14.1
|1.90
|66.82
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 8 mod 3 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|3,303,416
|3,249,427
|1.6%
|16.5
|0.91
|55.59
|style="text-align:left" |chr_LRUH 0 chr_H 0 MitM_CTL NG maxT 30000 NG_n 7 run
|-
|style="text-align:left" |Terry Ligocki
|3,249,427
|3,155,741
|2.9%
|13.4
|1.94
|67.25
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 6 mod 2 n 6 run
|-
|style="text-align:left" |Terry Ligocki
|3,155,741
|2,246,891
|28.8%
|14.1
|17.93
|62.24
|style="text-align:left" |chr_LRUH 8 chr_H 8 MitM_CTL NG maxT 30000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|2,246,891
|2,143,803
|4.6%
|7.5
|3.82
|83.19
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 3 n 1 run
|-
|style="text-align:left" |Terry Ligocki
|2,143,803
|1,938,663
|9.6%
|7.5
|7.58
|79.21
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 10000 LRUH 8 H 1 tH 1 n 4 run
|-
|style="text-align:left" |Terry Ligocki
|1,938,663
|1,885,153
|2.8%
|7.5
|1.98
|71.72
|style="text-align:left" |chr_LRUH 14 chr_H 12 MitM_CTL NG maxT 10000 NG_n 2 run
|-
|style="text-align:left" |Terry Ligocki
|1,885,153
|1,848,887
|1.9%
|10.5
|0.96
|49.68
|style="text-align:left" |MitM_CTL RWL_mod sim 1001 maxT 10000 H 3 mod 1 n 12 run
|-
|style="text-align:left" |Terry Ligocki
|1,848,887
|1,816,027
|1.8%
|7.6
|1.20
|67.77
|style="text-align:left" |chr_LRUH 18 chr_H 8 MitM_CTL NG maxT 10000 NG_n 5 run
|-
|style="text-align:left" |Terry Ligocki
|1,816,027
|1,688,951
|7.0%
|10.4
|3.40
|48.66
|style="text-align:left" |MitM_CTL CPS_LRU sim 1001 maxT 30000 LRUH 4 H 2 tH 0 n 2 run
|}
==References==


[[Category: BB Domains]]

BB(6)

2025-06-28T05:59:14Z

Peacemaker II:

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_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

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

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

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

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

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

Probviously non-halting Cryptids:

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

Probviously halting Cryptids:

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

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

Potential Cryptids:

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

== Top Halters ==
Below is a table of the machines with the 10 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of Knuth's up-arrow notation.<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE}}
|10 ↑↑ 15.60465
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA}}
|10 ↑↑ 7.23619
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB}}
|10 ↑↑ 6.96745
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE}}
|10 ↑↑ 5.77573
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD}}
|10 ↑↑ 5.63534
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC}}
|10 ↑↑ 5.56344
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

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

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

BB(6)

2025-06-28T05:58:36Z

Peacemaker II:

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_1RC---_1LD0RF_1RA0LE_0LD1RC_1RA0RE|halt}} was discovered by mxdys in June 2025, proving the lower bound:

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

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

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

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

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

Probviously non-halting Cryptids:

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

Probviously halting Cryptids:

* {{TM|1RB0RD_0RC1RE_1RD0LA_1LE1LC_1RF0LD_---0RA}}, [[Lucy's Moonlight]]
* {{TM|1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC|undecided}}, a family of 16 related TMs
* {{TM|1RB1RE_1LC1LD_---1LA_1LB1LE_0RF0RA_1LD1RF}}
* {{TM|1RB0RE_1LC1LD_0RA0LD_1LB0LA_1RF1RA_---1LB}}
* {{TM|1RB0LC_0LC0RF_1RD1LC_0RA1LE_---0LD_1LF1LA}}
* {{TM|1RB0LC_1LC0RD_1LF1LA_1LB1RE_1RB1LE_---0LE}}
* {{TM|1RB--- 0RC0RE 1RD1RF 1LE0LB 1RC0LD 1RC1RA}}
Although {{TM|1RB1LE_0LC0LB_1RD1LC_1RD1RA_1RF0LA_---1RE}} behaves similarly to the probviously halting Cryptids, it is estimated to have a 3/5 chance of becoming a [[translated cycler]] and a 2/5 chance of halting.

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

Potential Cryptids:

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

== Top Halters ==
Below is a table of the machines with the 10 highest known runtimes.<ref>Shawn Ligocki's list of 6-state, 2-symbol machines with large runtimes ([https://github.com/sligocki/busy-beaver/blob/main/Machines/bb/6x2.txt Link])</ref> Their sigma scores are expressed using an extension of Knuth's up-arrow notation.<ref>Shawn Ligocki. 2022. [https://www.sligocki.com/2022/06/25/ext-up-notation.html "Extending Up-arrow Notation"]</ref>
{| class="wikitable"
|+Top Known BB(6) Halters
!Standard format
!(approximate) Σ
|-
|{{TM|1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB|halt}}
|10 ↑↑ 11010000
|-
|{{TM|1RB0LD_1RC0RF_1LC1LA_0LE1RZ_1LF0RB_0RC0RE}}
|10 ↑↑ 15.60465
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LB0LE_1RZ0LC_1LA1LF}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1RF0LE_1RZ0LC_1LA1LF}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB0LF_1RC1RB_1LD0RA_1LF0LE_1RZ0LC_1LA1LF}}
|10 ↑↑ 7.52390
|-
|{{TM|1RB1RC_1LC1RE_1LD0LB_1RE1LC_1LE0RF_1RZ1RA}}
|10 ↑↑ 7.23619
|-
|{{TM|1RB1RA_1LC1LE_1RE0LD_1LC0LF_1RZ0RA_0RA0LB}}
|10 ↑↑ 6.96745
|-
|{{TM|1RB0RF_1LC0RA_1RZ0LD_1LE1LD_1RB1RC_0LD0RE}}
|10 ↑↑ 5.77573
|-
|{{TM|1RB0LA_1LC1LF_0LD0LC_0LE0LB_1RE0RA_1RZ1LD}}
|10 ↑↑ 5.63534
|-
|{{TM|1RB1RE_1LC1LF_1RD0LB_1LE0RC_1RA0LD_1RZ1LC}}
|10 ↑↑ 5.56344
|}
The runtimes are presumed to be about <math>\text{score}^2</math> which is roughly indistinguishable in tetration notation.

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

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

1RB--- 0RC0RE 1RD1RF 1LE0LB 1RC0LD 1RC1RA

2025-06-28T05:54:57Z

Peacemaker II:

{{machine|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}} is a [[probviously]] halting [[BB(6)]] [[Cryptid]] found by Racheline on 23 November 2024 ([https://discord.com/channels/960643023006490684/1239205785913790465/1309900024930635786 Discord link]). It has an interesting halting mechanism.

<pre>
`1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA` is a probviously halting cryptid. just an exponential one, specifically at each step it simply multiplies the main number by 8/3 and then changes a few things depending on what it was mod 3. the left side can change in a few ways to store information between these steps. if you start from the empty left side, there are some sequences of numbers you can get mod 3 which make the TM halt, and if the main number mod 3 was random, then the probability of this eventually happening would be 1 regardless of how long those sequences are. this is all standard stuff.

however
i'm pretty sure the shortest such sequence has length around 100. i think it's a fun puzzle to try to find one yourself. the shortest one i found on my own has length 110, though it's actually a group of 2^13\*3^15 = 117546246144 such sequences because some parts can be permuted with certain restrictions. specifically, each (n,m) in the following sequence of pairs can be replaced with any sequence of n 2's and m 0's whose first k elements contain at least k/3 2's for every k<=n+m:
||(1,0), (2,1), (1,0), (2,3), (1,0), (2,4), (1,0), (2,3), (1,0), (3,5), (1,0), (3,4), (1,0), (3,3), (1,0), (2,4), (1,0), (2,3), (1,0), (3,4), (1,0), (3,6), (1,0), (2,3), (1,0), (3,4), (1,0), (3,6), (1,0)|| (spoilered because i think it's a relatively fun puzzle to try to figure out on your own)
for any given point at which the left side is empty, the probability that it will follow one of these sequences from that point is just above 1/3^87
i also ran a search on all sequences of only 0's and 2's (1's seem to always reset most of the progress you made in getting to a halting config so i ignored those) up to that length, and the shortest it found was length 107, so the only way it's less than that would be if either my code is wrong or somehow 1's manage to be useful for something other than resetting.

so i think it's pretty reasonable to guess that from a given position with empty left side, the probability that the TM will halt before resetting the left side is around 1/3^80. the rest of the computation is the same as always: the expected number of times we generate a new random number mod 3 should be around 3^80, and since that happens only when the main number is roughly multiplied by 8/3, the number of 1s when halting should be around (8/3)^3^80 ≈ 10^10^37.8, and the halting time should be around (that)^2, i.e. around 10^10^38.1
i know, not that impressive compared to the current champion, but i'm posting it because i like the mechanism. also i guess it would still be in 11th place if this approximation is correct, though the current 11th place TM takes around 10^10^35.3 steps to halt so it's kinda close.
</pre>

[[Category:Cryptids]]

1RB--- 0RC0RE 1RD1RF 1LE0LB 1RC0LD 1RC1RA

2025-06-28T05:53:50Z

Peacemaker II: Created page with "{{machine|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}} {{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}} is a probviously halting BB(6) Cryptid found by Racheline on 23 November 2024 ([https://discord.com/channels/960643023006490684/1239205785913790465/1309900024930635786 Discord link]). <pre> `1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA` is a probviously halting cryptid. just an exponential one, specifically at each step it simply multiplies the main number by..."

{{machine|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}}
{{TM|1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA}} is a [[probviously]] halting [[BB(6)]] [[Cryptid]] found by Racheline on 23 November 2024 ([https://discord.com/channels/960643023006490684/1239205785913790465/1309900024930635786 Discord link]).

<pre>
`1RB---_0RC0RE_1RD1RF_1LE0LB_1RC0LD_1RC1RA` is a probviously halting cryptid. just an exponential one, specifically at each step it simply multiplies the main number by 8/3 and then changes a few things depending on what it was mod 3. the left side can change in a few ways to store information between these steps. if you start from the empty left side, there are some sequences of numbers you can get mod 3 which make the TM halt, and if the main number mod 3 was random, then the probability of this eventually happening would be 1 regardless of how long those sequences are. this is all standard stuff.

however
i'm pretty sure the shortest such sequence has length around 100. i think it's a fun puzzle to try to find one yourself. the shortest one i found on my own has length 110, though it's actually a group of 2^13\*3^15 = 117546246144 such sequences because some parts can be permuted with certain restrictions. specifically, each (n,m) in the following sequence of pairs can be replaced with any sequence of n 2's and m 0's whose first k elements contain at least k/3 2's for every k<=n+m:
||(1,0), (2,1), (1,0), (2,3), (1,0), (2,4), (1,0), (2,3), (1,0), (3,5), (1,0), (3,4), (1,0), (3,3), (1,0), (2,4), (1,0), (2,3), (1,0), (3,4), (1,0), (3,6), (1,0), (2,3), (1,0), (3,4), (1,0), (3,6), (1,0)|| (spoilered because i think it's a relatively fun puzzle to try to figure out on your own)
for any given point at which the left side is empty, the probability that it will follow one of these sequences from that point is just above 1/3^87
i also ran a search on all sequences of only 0's and 2's (1's seem to always reset most of the progress you made in getting to a halting config so i ignored those) up to that length, and the shortest it found was length 107, so the only way it's less than that would be if either my code is wrong or somehow 1's manage to be useful for something other than resetting.

so i think it's pretty reasonable to guess that from a given position with empty left side, the probability that the TM will halt before resetting the left side is around 1/3^80. the rest of the computation is the same as always: the expected number of times we generate a new random number mod 3 should be around 3^80, and since that happens only when the main number is roughly multiplied by 8/3, the number of 1s when halting should be around (8/3)^3^80 ≈ 10^10^37.8, and the halting time should be around (that)^2, i.e. around 10^10^38.1
i know, not that impressive compared to the current champion, but i'm posting it because i like the mechanism. also i guess it would still be in 11th place if this approximation is correct, though the current 11th place TM takes around 10^10^35.3 steps to halt so it's kinda close.
</pre>

[[Category:Cryptids]]

User talk:Mxdys

2025-06-17T06:11:56Z

Peacemaker II:

Are you really updated the lower bound of BB (6)? --[[User:Jacobzheng|Jacobzheng]] ([[User talk:Jacobzheng|talk]]) 05:48, 17 June 2025 (UTC)
:Mxdys recently discovered a new family of champion BB(6) machines and [https://discord.com/channels/960643023006490684/1384195691529633896 shared it on the Discord server]. A wiki article may be created soon. [[User:Peacemaker II|Peacemaker II]] ([[User talk:Peacemaker II|talk]]) 06:11, 17 June 2025 (UTC)

Wily Coyote

2025-06-15T20:11:08Z

Peacemaker II:

{{machine|1RB2LA1LA_2LA0RA2RC_---0LC2RA}}
{{TM|1RB2LA1LA_2LA0RA2RC_---0LC2RA}}

Also known as "Wily Coyote."
This is [[BB(3,3)]] holdout #531. It is equivalent to BB(3,3) holdout #532 {{TM|1RB2LA1LA_2LA0RA2RC_---1RB2RA}}, as found by [https://discord.com/channels/960643023006490684/1084047886494470185/1211902511829749810 @dyuan01].

@Legion wrote a simulator in Rust for this TM that can be found [https://discord.com/channels/960643023006490684/1026577255754903572/1244841134761578630 here.] They described the behavior of this machine as <blockquote>"The left side is just bouncing around a <code>1212</code>... pattern, nothing interesting there. But the right side has this extremely peculiar 'binary counter'. Since the head never moves more than a fixed number of cells (3, but effectively only 1) from right to left before finally resetting, we know that all information must flow from left to right. Therefore, it acts somewhat like a binary counter, in the sense that each cell must follow an eventually-periodic pattern as long as the machine doesn't halt, and that period is a non-decreasing power of 2. But as we look further to the right, the pattern actually gets more and more complex, as opposed to the trivial back-and-forth flipping of a real binary counter. The attached program calculates the pattern for each cell, as a function of the previous cell's pattern; by cell 99, the pattern is 11531 lines long. My current hope is in how repetitive the pattern appears; it often has just <code>BBBB</code>..., <code>CCCC</code>..., or <code>ADAD</code>... for hundreds of lines straight. So it might be bounded in how much information it actually contains, which would open a pathway for proving it non-halting."</blockquote>@Racheline wrote an accelerated simulator of this TM in Python:<syntaxhighlight lang="python" line="1">
def f():
global x, r, newr
for s in r[-1]:
i = 0
while s != 3:
if i >= len(x):
if s == 1:
print("halted")
exit()
elif s == 2:
x += [1, 1, 0, 0]
else:
x.append(0)
break
x[i], s = x[i] ^ (s >> 1) ^ 1, [1, 3, 2, 2, 0, 1][3 * x[i] + s]
i += 1
if i == 1:
newr.append(s)
if s == 3:
if i < len(x):
x[i] ^= 1
else:
x += [0, 0]

x = [0, 0]
y = [0, 0]
p = [1]
r = [[0, 1]]
for n in range(160):
newr = []
f()
if x[0] != y[0]:
f()
p.append(p[-1] + 1)
else:
p.append(p[-1])
r.append(newr)
if len(y) == 1:
newr = []
x = []
f()
y = [i for i in x]
else:
y.pop(0)
x = [i for i in y]
print(p[1:])
</syntaxhighlight>

This is Legion's "pet machine."

1LB--- 0RC1LE 0RD1RC 1LA1RB 0LB0LA

2025-05-21T00:26:44Z

Peacemaker II: Redirected page to Skelet 17

#REDIRECT [[Skelet 17]]

Skelet

2025-05-21T00:22:37Z

Peacemaker II: Undo revision 1985 by Peacemaker II (talk)

Georgi Ivanov Georgiev (aka '''Skelet''') was a Busy Beaver hunter who attempted to prove [[BB(5)]] in the early 2000s.

== bbfind ==
In 2003, he announced his results on his [https://skelet.ludost.net/bb/ webpage] in which he shared his code '''bbfind''' (a 6000 line Pascal program) which he claimed enumerated 150M 5-state TMs and reduced them down to 164 [[holdouts]] (he called them "nonregular machines"). After manually analyzing these 164 nonregular machines he reduced it to 43 holdouts which he called "hardly nonregular" (HNR) which were not solvable by his program nor by hand. These 43 HNR TMs became famous examples of hard TMs to prove and these TMs began to be named by their place on this list (such as [[Skelet 1]] and [[Skelet 17]]).

Due to the length and lack of comments in bbfind's source code, Skelet's work is not believed to have been independently verified.<ref>"[https://bbchallenge.org/story#skelets-43-undecided-machines Story]" (section "Skelet's 43 undecided machines").</ref> It is now known that all 43 of these holdouts do not halt. Other work done in bbfind includes computing the correct value of BB(4) and for "RTM(5)" (the set of all "[https://skelet.ludost.net/bb/RTM.htm Reversal Turing Machines]").

== Skelet's 43 HNR TMs ==

* Skelet's original list: https://skelet.ludost.net/bb/nreg.html
* [[bbchallenge.org]] list: https://bbchallenge.org/skelet (canonically numbers the TMs)
* More info on some of Skelet's 43 TMs: https://discuss.bbchallenge.org/t/proving-bb-5-in-coq/225/6
{| class="wikitable sortable"
|+
!Skelet #
!Skelet machine code
!Zoology
!Proof by [[Coq-BB5]]
|-
|1
|{{TM|1LC1LE_---1LD_1RD0LD_1LA1RE_0LB0RC}}
|[[Translated cycler]]<ref>https://www.sligocki.com/2023/03/13/skelet-1-infinite.html</ref>
|TABLE_BASED_SPORADIC
|-
|2
|{{TM|1LC0RE_---0RC_1RD0LA_1RA1RD_1LA0RB}}
|[[Translated cycler]]<ref name="tc">https://discuss.bbchallenge.org/t/skelet-machines-that-are-translated-cyclers/58</ref>
|TABLE_BASED_NG_3_6
|-
|3
|{{TM|1LC0RA_---1LE_1RD0LB_1RA1RC_0LC1LD}}
|
|TABLE_BASED_NG_3_6
|-
|4
|{{TM|1LC0RD_---0LE_1RD1LC_1LE1RA_1LB0LD}}
|
|TABLE_BASED_NG_3_2
|-
|5
|{{TM|1LC1LA_---0LD_1RD0LE_1LA0RC_1RC0LB}}
|[[Translated cycler]]<ref name="tc" />
|TABLE_BASED_NG_3_6
|-
|6
|{{TM|1LC0RB_---0RD_1LD0RA_1RE0LC_1RC1RE}}
|[[Translated cycler]]<ref name="tc" />
|TABLE_BASED_NG_3_6
|-
|7
|{{TM|1LC0RB_---1RE_1LD1LA_1RA0LD_0RA1RC}}
|
|TABLE_BASED_NG_3_6
|-
|8
|{{TM|1LC0RB_---0RC_1LD0LC_0RE1LC_0RA1RE}}
|
|TABLE_BASED_RWL_15_2
|-
|9
|{{TM|1LC1RD_---0LC_1RA1LC_1RE0RA_1LB0LE}}
|
|TABLE_BASED_DNV_12
|-
|10
|{{TM|1LC0LA_---0LC_0RD1LA_1LB1RE_1RD0RE}}
|Double Fibonacci counter<ref>https://www.sligocki.com/2023/03/14/skelet-10.html</ref>
|TABLE_BASED_SPORADIC
|-
|11
|{{TM|1LC0LA_---0RA_0RD1LA_0RE1RD_1LA0RB}}
|
|TABLE_BASED_RWL_15_2
|-
|12
|{{TM|1LC0LE_---1LE_0RD1LA_0LA1RC_1RC0LB}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|13
|{{TM|1LC0RB_---1RA_0LD1RE_0RE1LC_1RC0RA}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|14
|{{TM|1LB---_1RC0RE_1LD0RB_0LD1LA_0RC0LA}}
|[[Translated cycler]]<ref name="tc" />
|TABLE_BASED_Lp1
|-
|15
|{{TM|1LB---_1LC1RB_1RD1LE_1RB0RD_1LA0LC}}
|[[Shift overflow counter]]<ref name="so">https://www.sligocki.com/2023/02/05/shift-overflow.html</ref>
|TABLE_BASED_SPORADIC
|-
|16
|{{TM|1LB---_0RC1LD_1RD1RC_1LE0LE_0LA0RB}}
|
|TABLE_BASED_DNV_11
|-
|17
|{{TM|1LB---_0RC1LE_0RD1RC_1LA1RB_0LB0LA}}
|[[Dekaheptoid]]
|TABLE_BASED_SPORADIC
|-
|18
|{{TM|1LB---_0LC0RD_1LD0RE_1LE0LA_1RC0RD}}
|[[Translated cycler]]<ref name="tc" />
|TABLE_BASED_RWL_19_2
|-
|19
|{{TM|1LB---_0LC0LB_1RC0RD_1LA0RE_0RA0RE}}
|
|TABLE_BASED_DNV_10
|-
|20
|{{TM|1LB---_0LC1LD_0RD1LC_1RE0LA_1LA0RE}}
|[[Bouncer]]<ref name="ctl">https://discuss.bbchallenge.org/t/the-30-to-34-ctl-holdouts-from-bb-5/141</ref>
|TABLE_BASED_RWL_20_2
|-
|21
|{{TM|1LC1LE_1LA---_1RD0RE_1RB1RE_1RC0LA}}
|[[Bouncer]]<ref name="ctl" />
|TABLE_BASED_RWL_10_3
|-
|22
|{{TM|1LC0LE_1RA---_1RD0LA_0RD1RB_0LC0RB}}
|[[Translated cycler]]<ref name="tc" />
|TABLE_BASED_Lp1
|-
|23
|{{TM|1LC0RC_0LD---_1RD0LE_1LC0RE_1RA1LB}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|24
|{{TM|1LC1LA_1RE---_1RD0RD_0RB0LE_0LA1RC}}
|
|TABLE_BASED_DNV_10
|-
|25
|{{TM|1LC0RA_1LA---_1RD1LE_1RA0RD_0LE0RB}}
|[[Translated cycler]]<ref name="tc" />
|NGRAM_CPS_IMPL1_params_2_2_2_1600
|-
|26
|{{TM|1LC1RE_1RD---_1LD0LC_1RA1LD_1RB0RA}}
|[[Shift overflow counter]]<ref name="so" />
|TABLE_BASED_SPORADIC
|-
|27
|{{TM|1LC0RE_0LE---_1LD0LB_1RA0LA_0RA1RE}}
|
|NGRAM_CPS_IMPL1_params_2_3_3_1600
|-
|28
|{{TM|1LC0LE_1RD---_1LB1LE_1RA1RE_1LA0RD}}
|[[Bouncer]]<ref name="ctl" />
|TABLE_BASED_RWL_10_3
|-
|29
|{{TM|1LC0RD_0LA---_1RA0LD_1RE1LB_1LC0RC}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|30
|{{TM|1LC0LE_1RC---_0RD1LA_1RA0RE_1RB0LE}}
|
|TABLE_BASED_NG_6_11
|-
|31
|{{TM|1LC0RB_0RE---_0LD1LC_1LE0LC_1RA0RC}}
|
|TABLE_BASED_RWL_17_2
|-
|32
|{{TM|1LC0RE_0LC---_0LD0LB_1RD0RA_1RA1LD}}
|
|NGRAM_CPS_IMPL1_params_2_2_2_1600
|-
|33
|{{TM|1LC1RD_1RE---_0LD0LC_1RB0RA_1RA1LE}}
|[[Shift overflow counter]]<ref name="so" />
|TABLE_BASED_SPORADIC
|-
|34
|{{TM|1LC1RD_1RE---_0LD0LC_1RB0RA_1RA1LA}}
|[[Shift overflow counter]]<ref name="so" />
|TABLE_BASED_SPORADIC
|-
|35
|{{TM|1LC1RD_1RE---_0LD0LC_1RB0RA_1RA0RA}}
|[[Shift overflow counter]]<ref name="so" />
|TABLE_BASED_SPORADIC
|-
|36
|{{TM|1LC1RE_1RD---_0LD0LC_1RB1LA_1LD0RA}}
|
|TABLE_BASED_NG_2_2
|-
|37
|{{TM|1LC0RB_1RC---_0LD0RD_1RA0LE_1LD1LE}}
|
|NGRAM_CPS_IMPL1_params_2_2_2_1600
|-
|38
|{{TM|1LC0LC_1LD---_0LB0RD_0RE1LA_1RA1RE}}
|
|TABLE_BASED_DNV_10
|-
|39
|{{TM|1LB1LD_1RC---_1RE1RD_1LE0RC_1LA0LD}}
|[[Bouncer]]<ref name="ctl" />
|TABLE_BASED_RWL_10_3
|-
|40
|{{TM|1LB0LA_1RC---_0RC0RD_1LE0LB_0LE1LA}}
|
|TABLE_BASED_DNV_3
|-
|41
|{{TM|1LB0RA_1LC---_0LD1RE_1LE0LA_1RC0RA}}
|
|TABLE_BASED_NG_6_11
|-
|42
|{{TM|1LB0RE_1LC---_0LD0LC_1RD0RA_0RB0RE}}
|
|TABLE_BASED_DNV_9
|-
|43
|{{TM|1LB0RA_0LC---_1RC1LD_1LE1RA_0LB0RD}}
|
|TABLE_BASED_NG_7_10
|}

== References ==
<references/>

[[Category:Stub]]

Skelet

2025-05-21T00:20:30Z

Peacemaker II:

Georgi Ivanov Georgiev (aka '''Skelet''') was a Busy Beaver hunter who attempted to prove [[BB(5)]] in the early 2000s.

== bbfind ==
In 2003, he announced his results on his [https://skelet.ludost.net/bb/ webpage] in which he shared his code '''bbfind''' (a 6000 line Pascal program) which he claimed enumerated 150M 5-state TMs and reduced them down to 164 [[holdouts]] (he called them "nonregular machines"). After manually analyzing these 164 nonregular machines he reduced it to 43 holdouts which he called "hardly nonregular" (HNR) which were not solvable by his program nor by hand. These 43 HNR TMs became famous examples of hard TMs to prove and these TMs began to be named by their place on this list (such as [[Skelet 1]] and [[Skelet 17]]).

Due to the length and lack of comments in bbfind's source code, Skelet's work is not believed to have been independently verified.<ref>"[https://bbchallenge.org/story#skelets-43-undecided-machines Story]" (section "Skelet's 43 undecided machines").</ref> It is now known that all 43 of these holdouts do not halt. Other work done in bbfind includes computing the correct value of BB(4) and for "RTM(5)" (the set of all "[https://skelet.ludost.net/bb/RTM.htm Reversal Turing Machines]").

== Skelet's 43 HNR TMs ==

* Skelet's original list: https://skelet.ludost.net/bb/nreg.html
* [[bbchallenge.org]] list: https://bbchallenge.org/skelet (canonically numbers the TMs)
* More info on some of Skelet's 43 TMs: https://discuss.bbchallenge.org/t/proving-bb-5-in-coq/225/6
{| class="wikitable sortable"
|+
!Skelet #
!Skelet machine code
!Zoology
!Proof by [[Coq-BB5]]
|-
|[[Skelet 1|1]]
|{{TM|1LC1LE_---1LD_1RD0LD_1LA1RE_0LB0RC}}
|[[Translated cycler]]<ref>https://www.sligocki.com/2023/03/13/skelet-1-infinite.html</ref>
|TABLE_BASED_SPORADIC
|-
|2
|{{TM|1LC0RE_---0RC_1RD0LA_1RA1RD_1LA0RB}}
|[[Translated cycler]]<ref name="tc">https://discuss.bbchallenge.org/t/skelet-machines-that-are-translated-cyclers/58</ref>
|TABLE_BASED_NG_3_6
|-
|3
|{{TM|1LC0RA_---1LE_1RD0LB_1RA1RC_0LC1LD}}
|
|TABLE_BASED_NG_3_6
|-
|4
|{{TM|1LC0RD_---0LE_1RD1LC_1LE1RA_1LB0LD}}
|
|TABLE_BASED_NG_3_2
|-
|5
|{{TM|1LC1LA_---0LD_1RD0LE_1LA0RC_1RC0LB}}
|[[Translated cycler]]<ref name="tc" />
|TABLE_BASED_NG_3_6
|-
|6
|{{TM|1LC0RB_---0RD_1LD0RA_1RE0LC_1RC1RE}}
|[[Translated cycler]]<ref name="tc" />
|TABLE_BASED_NG_3_6
|-
|7
|{{TM|1LC0RB_---1RE_1LD1LA_1RA0LD_0RA1RC}}
|
|TABLE_BASED_NG_3_6
|-
|8
|{{TM|1LC0RB_---0RC_1LD0LC_0RE1LC_0RA1RE}}
|
|TABLE_BASED_RWL_15_2
|-
|9
|{{TM|1LC1RD_---0LC_1RA1LC_1RE0RA_1LB0LE}}
|
|TABLE_BASED_DNV_12
|-
|10
|{{TM|1LC0LA_---0LC_0RD1LA_1LB1RE_1RD0RE}}
|Double Fibonacci counter<ref>https://www.sligocki.com/2023/03/14/skelet-10.html</ref>
|TABLE_BASED_SPORADIC
|-
|11
|{{TM|1LC0LA_---0RA_0RD1LA_0RE1RD_1LA0RB}}
|
|TABLE_BASED_RWL_15_2
|-
|12
|{{TM|1LC0LE_---1LE_0RD1LA_0LA1RC_1RC0LB}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|13
|{{TM|1LC0RB_---1RA_0LD1RE_0RE1LC_1RC0RA}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|14
|{{TM|1LB---_1RC0RE_1LD0RB_0LD1LA_0RC0LA}}
|[[Translated cycler]]<ref name="tc" />
|TABLE_BASED_Lp1
|-
|15
|{{TM|1LB---_1LC1RB_1RD1LE_1RB0RD_1LA0LC}}
|[[Shift overflow counter]]<ref name="so">https://www.sligocki.com/2023/02/05/shift-overflow.html</ref>
|TABLE_BASED_SPORADIC
|-
|16
|{{TM|1LB---_0RC1LD_1RD1RC_1LE0LE_0LA0RB}}
|
|TABLE_BASED_DNV_11
|-
|[[Skelet 17|17]]
|{{TM|1LB---_0RC1LE_0RD1RC_1LA1RB_0LB0LA}}
|[[Dekaheptoid]]
|TABLE_BASED_SPORADIC
|-
|18
|{{TM|1LB---_0LC0RD_1LD0RE_1LE0LA_1RC0RD}}
|[[Translated cycler]]<ref name="tc" />
|TABLE_BASED_RWL_19_2
|-
|19
|{{TM|1LB---_0LC0LB_1RC0RD_1LA0RE_0RA0RE}}
|
|TABLE_BASED_DNV_10
|-
|20
|{{TM|1LB---_0LC1LD_0RD1LC_1RE0LA_1LA0RE}}
|[[Bouncer]]<ref name="ctl">https://discuss.bbchallenge.org/t/the-30-to-34-ctl-holdouts-from-bb-5/141</ref>
|TABLE_BASED_RWL_20_2
|-
|21
|{{TM|1LC1LE_1LA---_1RD0RE_1RB1RE_1RC0LA}}
|[[Bouncer]]<ref name="ctl" />
|TABLE_BASED_RWL_10_3
|-
|22
|{{TM|1LC0LE_1RA---_1RD0LA_0RD1RB_0LC0RB}}
|[[Translated cycler]]<ref name="tc" />
|TABLE_BASED_Lp1
|-
|23
|{{TM|1LC0RC_0LD---_1RD0LE_1LC0RE_1RA1LB}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|24
|{{TM|1LC1LA_1RE---_1RD0RD_0RB0LE_0LA1RC}}
|
|TABLE_BASED_DNV_10
|-
|25
|{{TM|1LC0RA_1LA---_1RD1LE_1RA0RD_0LE0RB}}
|[[Translated cycler]]<ref name="tc" />
|NGRAM_CPS_IMPL1_params_2_2_2_1600
|-
|26
|{{TM|1LC1RE_1RD---_1LD0LC_1RA1LD_1RB0RA}}
|[[Shift overflow counter]]<ref name="so" />
|TABLE_BASED_SPORADIC
|-
|27
|{{TM|1LC0RE_0LE---_1LD0LB_1RA0LA_0RA1RE}}
|
|NGRAM_CPS_IMPL1_params_2_3_3_1600
|-
|28
|{{TM|1LC0LE_1RD---_1LB1LE_1RA1RE_1LA0RD}}
|[[Bouncer]]<ref name="ctl" />
|TABLE_BASED_RWL_10_3
|-
|29
|{{TM|1LC0RD_0LA---_1RA0LD_1RE1LB_1LC0RC}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|30
|{{TM|1LC0LE_1RC---_0RD1LA_1RA0RE_1RB0LE}}
|
|TABLE_BASED_NG_6_11
|-
|31
|{{TM|1LC0RB_0RE---_0LD1LC_1LE0LC_1RA0RC}}
|
|TABLE_BASED_RWL_17_2
|-
|32
|{{TM|1LC0RE_0LC---_0LD0LB_1RD0RA_1RA1LD}}
|
|NGRAM_CPS_IMPL1_params_2_2_2_1600
|-
|33
|{{TM|1LC1RD_1RE---_0LD0LC_1RB0RA_1RA1LE}}
|[[Shift overflow counter]]<ref name="so" />
|TABLE_BASED_SPORADIC
|-
|34
|{{TM|1LC1RD_1RE---_0LD0LC_1RB0RA_1RA1LA}}
|[[Shift overflow counter]]<ref name="so" />
|TABLE_BASED_SPORADIC
|-
|35
|{{TM|1LC1RD_1RE---_0LD0LC_1RB0RA_1RA0RA}}
|[[Shift overflow counter]]<ref name="so" />
|TABLE_BASED_SPORADIC
|-
|36
|{{TM|1LC1RE_1RD---_0LD0LC_1RB1LA_1LD0RA}}
|
|TABLE_BASED_NG_2_2
|-
|37
|{{TM|1LC0RB_1RC---_0LD0RD_1RA0LE_1LD1LE}}
|
|NGRAM_CPS_IMPL1_params_2_2_2_1600
|-
|38
|{{TM|1LC0LC_1LD---_0LB0RD_0RE1LA_1RA1RE}}
|
|TABLE_BASED_DNV_10
|-
|39
|{{TM|1LB1LD_1RC---_1RE1RD_1LE0RC_1LA0LD}}
|[[Bouncer]]<ref name="ctl" />
|TABLE_BASED_RWL_10_3
|-
|40
|{{TM|1LB0LA_1RC---_0RC0RD_1LE0LB_0LE1LA}}
|
|TABLE_BASED_DNV_3
|-
|41
|{{TM|1LB0RA_1LC---_0LD1RE_1LE0LA_1RC0RA}}
|
|TABLE_BASED_NG_6_11
|-
|42
|{{TM|1LB0RE_1LC---_0LD0LC_1RD0RA_0RB0RE}}
|
|TABLE_BASED_DNV_9
|-
|43
|{{TM|1LB0RA_0LC---_1RC1LD_1LE1RA_0LB0RD}}
|
|TABLE_BASED_NG_7_10
|}

== References ==
<references/>

[[Category:Stub]]

Skelet

2025-05-21T00:02:31Z

Peacemaker II:

Skelet

2025-05-20T22:46:24Z

Peacemaker II: added refs

Georgi Ivanov Georgiev (aka '''Skelet''') was a Busy Beaver hunter who attempted to prove [[BB(5)]] in the early 2000s.

== bbfind ==
In 2003, he announced his results on his [https://skelet.ludost.net/bb/ webpage] in which he shared his code '''bbfind''' (a 6000 line Pascal program) which he claimed enumerated 150M 5-state TMs and reduced them down to 164 [[holdouts]] (he called them "nonregular machines"). After manually analyzing these 164 nonregular machines he reduced it to 43 holdouts which he called "hardly nonregular" (HNR) which were not solvable by his program nor by hand. These 43 HNR TMs became famous examples of hard TMs to prove and these TMs began to be named by their place on this list (such as [[Skelet 1]] and [[Skelet 17]]).

Due to the length and lack of comments in bbfind's source code, Skelet's work is not believed to have been independently verified.<ref>"[https://bbchallenge.org/story#skelets-43-undecided-machines Story]" (section "Skelet's 43 undecided machines").</ref> It is now known that all 43 of these holdouts do not halt. Other work done in bbfind includes computing the correct value of BB(4) and for "RTM(5)" (the set of all "[https://skelet.ludost.net/bb/RTM.htm Reversal Turing Machines]").

== Skelet's 43 HNR TMs ==

* Skelet's original list: https://skelet.ludost.net/bb/nreg.html
* [[bbchallenge.org]] list: https://bbchallenge.org/skelet (canonically numbers the TMs)
* More info on some of Skelet's 43 TMs: https://discuss.bbchallenge.org/t/proving-bb-5-in-coq/225/6
{| class="wikitable sortable"
|+
!Skelet #
!Skelet machine code
!Zoology
!Proof by [[Coq-BB5]]
|-
|1
|{{TM|1LC1LE_---1LD_1RD0LD_1LA1RE_0LB0RC}}
|[[Translated cycler]]<ref>https://www.sligocki.com/2023/03/13/skelet-1-infinite.html</ref>
|TABLE_BASED_SPORADIC
|-
|2
|{{TM|1LC0RE_---0RC_1RD0LA_1RA1RD_1LA0RB}}
|
|TABLE_BASED_NG_3_6
|-
|3
|{{TM|1LC0RA_---1LE_1RD0LB_1RA1RC_0LC1LD}}
|
|TABLE_BASED_NG_3_6
|-
|4
|{{TM|1LC0RD_---0LE_1RD1LC_1LE1RA_1LB0LD}}
|
|TABLE_BASED_NG_3_2
|-
|5
|{{TM|1LC1LA_---0LD_1RD0LE_1LA0RC_1RC0LB}}
|
|TABLE_BASED_NG_3_6
|-
|6
|{{TM|1LC0RB_---0RD_1LD0RA_1RE0LC_1RC1RE}}
|
|TABLE_BASED_NG_3_6
|-
|7
|{{TM|1LC0RB_---1RE_1LD1LA_1RA0LD_0RA1RC}}
|
|TABLE_BASED_NG_3_6
|-
|8
|{{TM|1LC0RB_---0RC_1LD0LC_0RE1LC_0RA1RE}}
|
|TABLE_BASED_RWL_15_2
|-
|9
|{{TM|1LC1RD_---0LC_1RA1LC_1RE0RA_1LB0LE}}
|
|TABLE_BASED_DNV_12
|-
|10
|{{TM|1LC0LA_---0LC_0RD1LA_1LB1RE_1RD0RE}}
|Double Fibonacci counter<ref>https://www.sligocki.com/2023/03/14/skelet-10.html</ref>
|TABLE_BASED_SPORADIC
|-
|11
|{{TM|1LC0LA_---0RA_0RD1LA_0RE1RD_1LA0RB}}
|
|TABLE_BASED_RWL_15_2
|-
|12
|{{TM|1LC0LE_---1LE_0RD1LA_0LA1RC_1RC0LB}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|13
|{{TM|1LC0RB_---1RA_0LD1RE_0RE1LC_1RC0RA}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|14
|{{TM|1LB---_1RC0RE_1LD0RB_0LD1LA_0RC0LA}}
|
|TABLE_BASED_Lp1
|-
|15
|{{TM|1LB---_1LC1RB_1RD1LE_1RB0RD_1LA0LC}}
|[[Shift overflow counter]]<ref name="so">https://www.sligocki.com/2023/02/05/shift-overflow.html</ref>
|TABLE_BASED_SPORADIC
|-
|16
|{{TM|1LB---_0RC1LD_1RD1RC_1LE0LE_0LA0RB}}
|
|TABLE_BASED_DNV_11
|-
|17
|{{TM|1LB---_0RC1LE_0RD1RC_1LA1RB_0LB0LA}}
|[[Dekaheptoid]]
|TABLE_BASED_SPORADIC
|-
|18
|{{TM|1LB---_0LC0RD_1LD0RE_1LE0LA_1RC0RD}}
|
|TABLE_BASED_RWL_19_2
|-
|19
|{{TM|1LB---_0LC0LB_1RC0RD_1LA0RE_0RA0RE}}
|
|TABLE_BASED_DNV_10
|-
|20
|{{TM|1LB---_0LC1LD_0RD1LC_1RE0LA_1LA0RE}}
|
|TABLE_BASED_RWL_20_2
|-
|21
|{{TM|1LC1LE_1LA---_1RD0RE_1RB1RE_1RC0LA}}
|
|TABLE_BASED_RWL_10_3
|-
|22
|{{TM|1LC0LE_1RA---_1RD0LA_0RD1RB_0LC0RB}}
|
|TABLE_BASED_Lp1
|-
|23
|{{TM|1LC0RC_0LD---_1RD0LE_1LC0RE_1RA1LB}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|24
|{{TM|1LC1LA_1RE---_1RD0RD_0RB0LE_0LA1RC}}
|
|TABLE_BASED_DNV_10
|-
|25
|{{TM|1LC0RA_1LA---_1RD1LE_1RA0RD_0LE0RB}}
|
|NGRAM_CPS_IMPL1_params_2_2_2_1600
|-
|26
|{{TM|1LC1RE_1RD---_1LD0LC_1RA1LD_1RB0RA}}
|[[Shift overflow counter]]<ref name="so" />
|TABLE_BASED_SPORADIC
|-
|27
|{{TM|1LC0RE_0LE---_1LD0LB_1RA0LA_0RA1RE}}
|
|NGRAM_CPS_IMPL1_params_2_3_3_1600
|-
|28
|{{TM|1LC0LE_1RD---_1LB1LE_1RA1RE_1LA0RD}}
|
|TABLE_BASED_RWL_10_3
|-
|29
|{{TM|1LC0RD_0LA---_1RA0LD_1RE1LB_1LC0RC}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|30
|{{TM|1LC0LE_1RC---_0RD1LA_1RA0RE_1RB0LE}}
|
|TABLE_BASED_NG_6_11
|-
|31
|{{TM|1LC0RB_0RE---_0LD1LC_1LE0LC_1RA0RC}}
|
|TABLE_BASED_RWL_17_2
|-
|32
|{{TM|1LC0RE_0LC---_0LD0LB_1RD0RA_1RA1LD}}
|
|NGRAM_CPS_IMPL1_params_2_2_2_1600
|-
|33
|{{TM|1LC1RD_1RE---_0LD0LC_1RB0RA_1RA1LE}}
|[[Shift overflow counter]]<ref name="so" />
|TABLE_BASED_SPORADIC
|-
|34
|{{TM|1LC1RD_1RE---_0LD0LC_1RB0RA_1RA1LA}}
|[[Shift overflow counter]]<ref name="so" />
|TABLE_BASED_SPORADIC
|-
|35
|{{TM|1LC1RD_1RE---_0LD0LC_1RB0RA_1RA0RA}}
|[[Shift overflow counter]]<ref name="so" />
|TABLE_BASED_SPORADIC
|-
|36
|{{TM|1LC1RE_1RD---_0LD0LC_1RB1LA_1LD0RA}}
|
|TABLE_BASED_NG_2_2
|-
|37
|{{TM|1LC0RB_1RC---_0LD0RD_1RA0LE_1LD1LE}}
|
|NGRAM_CPS_IMPL1_params_2_2_2_1600
|-
|38
|{{TM|1LC0LC_1LD---_0LB0RD_0RE1LA_1RA1RE}}
|
|TABLE_BASED_DNV_10
|-
|39
|{{TM|1LB1LD_1RC---_1RE1RD_1LE0RC_1LA0LD}}
|
|TABLE_BASED_RWL_10_3
|-
|40
|{{TM|1LB0LA_1RC---_0RC0RD_1LE0LB_0LE1LA}}
|
|TABLE_BASED_DNV_3
|-
|41
|{{TM|1LB0RA_1LC---_0LD1RE_1LE0LA_1RC0RA}}
|
|TABLE_BASED_NG_6_11
|-
|42
|{{TM|1LB0RE_1LC---_0LD0LC_1RD0RA_0RB0RE}}
|
|TABLE_BASED_DNV_9
|-
|43
|{{TM|1LB0RA_0LC---_1RC1LD_1LE1RA_0LB0RD}}
|
|TABLE_BASED_NG_7_10
|}

== References ==
<references/>

[[Category:Stub]]

Skelet

2025-05-20T22:28:51Z

Peacemaker II: /* Skelet's 43 HNR TMs */

Georgi Ivanov Georgiev (aka '''Skelet''') was a Busy Beaver hunter who attempted to prove [[BB(5)]] in the early 2000s.

== bbfind ==
In 2003, he announced his results on his [https://skelet.ludost.net/bb/ webpage] in which he shared his code '''bbfind''' (a 6000 line Pascal program) which he claimed enumerated 150M 5-state TMs and reduced them down to 164 [[holdouts]] (he called them "nonregular machines"). After manually analyzing these 164 nonregular machines he reduced it to 43 holdouts which he called "hardly nonregular" (HNR) which were not solvable by his program nor by hand. These 43 HNR TMs became famous examples of hard TMs to prove and these TMs began to be named by their place on this list (such as [[Skelet 1]] and [[Skelet 17]]).

Due to the length and lack of comments in bbfind's source code, Skelet's work is not believed to have been independently verified.<ref>"[https://bbchallenge.org/story#skelets-43-undecided-machines Story]" (section "Skelet's 43 undecided machines").</ref> It is now known that all 43 of these holdouts do not halt. Other work done in bbfind includes computing the correct value of BB(4) and for "RTM(5)" (the set of all "[https://skelet.ludost.net/bb/RTM.htm Reversal Turing Machines]").

== Skelet's 43 HNR TMs ==

* Skelet's original list: https://skelet.ludost.net/bb/nreg.html
* [[bbchallenge.org]] list: https://bbchallenge.org/skelet (canonically numbers the TMs)
* More info on some of Skelet's 43 TMs: https://discuss.bbchallenge.org/t/proving-bb-5-in-coq/225/6
{| class="wikitable sortable"
|+
!Skelet #
!Skelet machine code
!Zoology
!Proof by [[Coq-BB5]]
|-
|1
|{{TM|1LC1LE_---1LD_1RD0LD_1LA1RE_0LB0RC}}
|[[Translated cycler]]
|TABLE_BASED_SPORADIC
|-
|2
|{{TM|1LC0RE_---0RC_1RD0LA_1RA1RD_1LA0RB}}
|
|TABLE_BASED_NG_3_6
|-
|3
|{{TM|1LC0RA_---1LE_1RD0LB_1RA1RC_0LC1LD}}
|
|TABLE_BASED_NG_3_6
|-
|4
|{{TM|1LC0RD_---0LE_1RD1LC_1LE1RA_1LB0LD}}
|
|TABLE_BASED_NG_3_2
|-
|5
|{{TM|1LC1LA_---0LD_1RD0LE_1LA0RC_1RC0LB}}
|
|TABLE_BASED_NG_3_6
|-
|6
|{{TM|1LC0RB_---0RD_1LD0RA_1RE0LC_1RC1RE}}
|
|TABLE_BASED_NG_3_6
|-
|7
|{{TM|1LC0RB_---1RE_1LD1LA_1RA0LD_0RA1RC}}
|
|TABLE_BASED_NG_3_6
|-
|8
|{{TM|1LC0RB_---0RC_1LD0LC_0RE1LC_0RA1RE}}
|
|TABLE_BASED_RWL_15_2
|-
|9
|{{TM|1LC1RD_---0LC_1RA1LC_1RE0RA_1LB0LE}}
|
|TABLE_BASED_DNV_12
|-
|10
|{{TM|1LC0LA_---0LC_0RD1LA_1LB1RE_1RD0RE}}
|Double Fibonacci Counter
|TABLE_BASED_SPORADIC
|-
|11
|{{TM|1LC0LA_---0RA_0RD1LA_0RE1RD_1LA0RB}}
|
|TABLE_BASED_RWL_15_2
|-
|12
|{{TM|1LC0LE_---1LE_0RD1LA_0LA1RC_1RC0LB}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|13
|{{TM|1LC0RB_---1RA_0LD1RE_0RE1LC_1RC0RA}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|14
|{{TM|1LB---_1RC0RE_1LD0RB_0LD1LA_0RC0LA}}
|
|TABLE_BASED_Lp1
|-
|15
|{{TM|1LB---_1LC1RB_1RD1LE_1RB0RD_1LA0LC}}
|[[Shift overflow counter]]
|TABLE_BASED_SPORADIC
|-
|16
|{{TM|1LB---_0RC1LD_1RD1RC_1LE0LE_0LA0RB}}
|
|TABLE_BASED_DNV_11
|-
|17
|{{TM|1LB---_0RC1LE_0RD1RC_1LA1RB_0LB0LA}}
|[[Dekaheptoid]]
|TABLE_BASED_SPORADIC
|-
|18
|{{TM|1LB---_0LC0RD_1LD0RE_1LE0LA_1RC0RD}}
|
|TABLE_BASED_RWL_19_2
|-
|19
|{{TM|1LB---_0LC0LB_1RC0RD_1LA0RE_0RA0RE}}
|
|TABLE_BASED_DNV_10
|-
|20
|{{TM|1LB---_0LC1LD_0RD1LC_1RE0LA_1LA0RE}}
|
|TABLE_BASED_RWL_20_2
|-
|21
|{{TM|1LC1LE_1LA---_1RD0RE_1RB1RE_1RC0LA}}
|
|TABLE_BASED_RWL_10_3
|-
|22
|{{TM|1LC0LE_1RA---_1RD0LA_0RD1RB_0LC0RB}}
|
|TABLE_BASED_Lp1
|-
|23
|{{TM|1LC0RC_0LD---_1RD0LE_1LC0RE_1RA1LB}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|24
|{{TM|1LC1LA_1RE---_1RD0RD_0RB0LE_0LA1RC}}
|
|TABLE_BASED_DNV_10
|-
|25
|{{TM|1LC0RA_1LA---_1RD1LE_1RA0RD_0LE0RB}}
|
|NGRAM_CPS_IMPL1_params_2_2_2_1600
|-
|26
|{{TM|1LC1RE_1RD---_1LD0LC_1RA1LD_1RB0RA}}
|[[Shift overflow counter]]
|TABLE_BASED_SPORADIC
|-
|27
|{{TM|1LC0RE_0LE---_1LD0LB_1RA0LA_0RA1RE}}
|
|NGRAM_CPS_IMPL1_params_2_3_3_1600
|-
|28
|{{TM|1LC0LE_1RD---_1LB1LE_1RA1RE_1LA0RD}}
|
|TABLE_BASED_RWL_10_3
|-
|29
|{{TM|1LC0RD_0LA---_1RA0LD_1RE1LB_1LC0RC}}
|
|NGRAM_CPS_IMPL1_params_6_2_2_3200
|-
|30
|{{TM|1LC0LE_1RC---_0RD1LA_1RA0RE_1RB0LE}}
|
|TABLE_BASED_NG_6_11
|-
|31
|{{TM|1LC0RB_0RE---_0LD1LC_1LE0LC_1RA0RC}}
|
|TABLE_BASED_RWL_17_2
|-
|32
|{{TM|1LC0RE_0LC---_0LD0LB_1RD0RA_1RA1LD}}
|
|NGRAM_CPS_IMPL1_params_2_2_2_1600
|-
|33
|{{TM|1LC1RD_1RE---_0LD0LC_1RB0RA_1RA1LE}}
|[[Shift overflow counter]]
|TABLE_BASED_SPORADIC
|-
|34
|{{TM|1LC1RD_1RE---_0LD0LC_1RB0RA_1RA1LA}}
|[[Shift overflow counter]]
|TABLE_BASED_SPORADIC
|-
|35
|{{TM|1LC1RD_1RE---_0LD0LC_1RB0RA_1RA0RA}}
|[[Shift overflow counter]]
|TABLE_BASED_SPORADIC
|-
|36
|{{TM|1LC1RE_1RD---_0LD0LC_1RB1LA_1LD0RA}}
|
|TABLE_BASED_NG_2_2
|-
|37
|{{TM|1LC0RB_1RC---_0LD0RD_1RA0LE_1LD1LE}}
|
|NGRAM_CPS_IMPL1_params_2_2_2_1600
|-
|38
|{{TM|1LC0LC_1LD---_0LB0RD_0RE1LA_1RA1RE}}
|
|TABLE_BASED_DNV_10
|-
|39
|{{TM|1LB1LD_1RC---_1RE1RD_1LE0RC_1LA0LD}}
|
|TABLE_BASED_RWL_10_3
|-
|40
|{{TM|1LB0LA_1RC---_0RC0RD_1LE0LB_0LE1LA}}
|
|TABLE_BASED_DNV_3
|-
|41
|{{TM|1LB0RA_1LC---_0LD1RE_1LE0LA_1RC0RA}}
|
|TABLE_BASED_NG_6_11
|-
|42
|{{TM|1LB0RE_1LC---_0LD0LC_1RD0RA_0RB0RE}}
|
|TABLE_BASED_DNV_9
|-
|43
|{{TM|1LB0RA_0LC---_1RC1LD_1LE1RA_0LB0RD}}
|
|TABLE_BASED_NG_7_10
|}

== References ==
<references/>

[[Category:Stub]]

User:MrSolis

2025-04-30T16:19:05Z

Peacemaker II: /* Discord Server */

== Discord Server ==
Hi MrSolis,
I like your contributions to the Hydra and Antihydra articles. Do you mind joining the bbchallenge discord server? It's easier to have a conversation there. https://discord.com/invite/3uqtPJA9Uv [[User:Peacemaker II|Peacemaker II]] ([[User talk:Peacemaker II|talk]]) 15:58, 30 April 2025 (UTC)

:I dont have a disc account and also dont feel like conversing rn but will prob join today/tmrw [[User:MrSolis|MrSolis]] ([[User:MrSolis|talk]]) 16:09, 30 April 2025 (UTC)

:: 👍 [[User:Peacemaker II|Peacemaker II]] ([[User talk:Peacemaker II|talk]]) 16:19, 30 April 2025 (UTC)

User:MrSolis

2025-04-30T15:58:18Z

Peacemaker II: Created page with "== Discord Server == Hi MrSolis, I like your contributions to the Hydra and Antihydra articles. Do you mind joining the bbchallenge discord server? It's easier to have a conversation there. https://discord.com/invite/3uqtPJA9Uv ~~~~"

Talk:Antihydra

2025-04-30T00:36:59Z

Peacemaker II: Created page with "== Variable Naming Conventions == I noticed that the usage of the variables 'a' and 'b' in this article are opposite to how they are defined in other sources. Can we modify this article to make it more consistent? ~~~~"

== Variable Naming Conventions ==
I noticed that the usage of the variables 'a' and 'b' in this article are opposite to how they are defined in other sources. Can we modify this article to make it more consistent? [[User:Peacemaker II|Peacemaker II]] ([[User talk:Peacemaker II|talk]]) 00:36, 30 April 2025 (UTC)

BB(2,5)

2025-03-29T17:43:46Z

Peacemaker II: /* Holdouts */ updated holdout count

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

The current BB(2,5) champion is {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}, also discovered by Daniel Yuan in June 2024. It is a halting [[shift overflow counter]], and is in fact the only known champion machine that exhibits [[Counter]] behavior. It provides the lower bound:

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

== Cryptids ==
Known BB(2,5) Cryptids

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA}}, known as [[Hydra]].
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}, known as the bonus cryptid.

== Holdouts ==
In April 2024, Shawn Ligocki publicly released a list of 23,411 undecided BB(2,5) machines. Justin Blanchard then made substantial progress over the course of the next month, reducing the list to 499 holdouts by late May 2024. In June 2024, @mxdys cut down the list to 273 using halting and inductive deciders, and again to 217 using [[Closed Tape Language (CTL)|CTL]]. In February 2025, @mxdys ran a decider pipeline in Coq that resulted in only 173 holdouts. Since then, additional machines have been proven in Coq using both deciders and individual proofs. As of March 2025, there are only 83 machines that do not have a proof in Coq. These machines have all been run for at least 1 trillion steps [https://discord.com/channels/960643023006490684/1259770421046411285/1354323123293655111 without halting by mxdys.]

[[Category:BB Domain]]

BB(2,5)

2025-03-26T06:28:45Z

Peacemaker II: /* Holdouts */

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

The current BB(2,5) champion is {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}, also discovered by Daniel Yuan in June 2024. It is a halting [[shift overflow counter]], and is in fact the only known champion machine that exhibits [[Counter]] behavior. It provides the lower bound:

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

== Cryptids ==
Known BB(2,5) Cryptids

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA}}, known as [[Hydra]].
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}, known as the bonus cryptid.

== Holdouts ==
In April 2024, Shawn Ligocki publicly released a list of 23,411 undecided BB(2,5) machines. Justin Blanchard then made substantial progress over the course of the next month, reducing the list to 499 holdouts by late May 2024. In June 2024, @mxdys cut down the list to 273 using halting and inductive deciders, and again to 217 using [[Closed Tape Language (CTL)|CTL]]. In February 2025, @mxdys ran a decider pipeline in Coq that resulted in only 173 holdouts. Since then, additional machines have been proven in Coq using both deciders and individual proofs. As of March 2025, there are only 99 machines that do not have a proof in Coq. These machines have all been run for at least 1 trillion steps without halting.

[[Category:BB Domain]]

BB(2,5)

2025-03-19T19:07:27Z

Peacemaker II: /* Holdouts */ updated info

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

The current BB(2,5) champion is {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}, also discovered by Daniel Yuan in June 2024. It is a halting [[shift overflow counter]], and is in fact the only known champion machine that exhibits [[Counter]] behavior. It provides the lower bound:

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

== Cryptids ==
Known BB(2,5) Cryptids

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA}}, known as [[Hydra]].
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}, known as the bonus cryptid.

== Holdouts ==
In April 2024, Shawn Ligocki publicly released a list of 23,411 undecided BB(2,5) machines. Justin Blanchard then made substantial progress over the course of the next month, reducing the list to 499 holdouts by late May 2024. In June 2024, @mxdys cut down the list to 273 using halting and inductive deciders, and again to 217 using [[Closed Tape Language (CTL)|CTL]]. This list of holdouts can be found [[:File:2x5 holdouts 217.txt|here]].

In February 2025, @mxdys ran a decider pipeline in Coq that resulted in only 173 holdouts. Since then, additional machines have been proven in Coq using both deciders and individual proofs. As of March 2025, there are only 99 BB(2,5) machines that do not have a proof in Coq.

[[Category:BB Domain]]

BB(2,5)

2025-02-27T23:59:42Z

Peacemaker II: /* Holdouts */ clarified wording

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

The current BB(2,5) champion is {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}, also discovered by Daniel Yuan in June 2024. It is a halting [[shift overflow counter]], and is in fact the only known champion machine that exhibits [[Counter]] behavior. It provides the lower bound:

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

== Cryptids ==
Known BB(2,5) Cryptids

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA}}, known as [[Hydra]].
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}, known as the bonus cryptid.

== Holdouts ==
In April 2024, Shawn Ligocki publicly released a list of 23,411 undecided BB(2,5) machines. Justin Blanchard then made substantial progress over the course of the next month, reducing the list to 499 holdouts by late May 2024. In June 2024, @mxdys cut down the list to 273 using halting and inductive deciders, and again to 217 using [[Closed Tape Language|CTL]]. This list of holdouts can be found [[:File:2x5 holdouts 217.txt|here]].

In February 2025, @mxdys ran a decider pipeline in [[Coq]] that resulted in only 173 holdouts. This list is not a subset of previous lists due to 20 machines previously solved by Justin Blanchard not yet being implemented in Coq. Therefore, the actual list of holdouts as of February 2025 is as low as 153.

[[Category:BB Domain]]

BB(2,5)

2025-02-27T19:30:54Z

Peacemaker II: /* Holdouts */

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

The current BB(2,5) champion is {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}, also discovered by Daniel Yuan in June 2024. It is a halting [[shift overflow counter]], and is in fact the only known champion machine that exhibits [[Counter]] behavior. It provides the lower bound:

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

== Cryptids ==
Known BB(2,5) Cryptids

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA}}, known as [[Hydra]].
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}, known as the bonus cryptid.

== Holdouts ==
In April 2024, Shawn Ligocki publicly released a list of 23,411 undecided BB(2,5) machines. Justin Blanchard then made substantial progress over the course of the next month, reducing the list to 499 holdouts by late May 2024. In June 2024, @mxdys cut down the list to 273 using halting and inductive deciders, and again to 217 using [[Closed Tape Language|CTL]]. This list of holdouts can be found [[:File:2x5 holdouts 217.txt|here]].

In February 2025, @mxdys ran a decider pipeline in [[Coq]] that resulted in only 173 holdouts. This list has some overlap with previous lists due to 20 machines previously solved by Justin Blanchard not yet being implemented in Coq. Therefore, the list of holdouts as of February 2025 is as low as 153.

[[Category:BB Domain]]

BB(2,5)

2025-02-27T19:29:41Z

Peacemaker II: /* Holdouts */ Updated holdout list progress and info

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

The current BB(2,5) champion is {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}, also discovered by Daniel Yuan in June 2024. It is a halting [[shift overflow counter]], and is in fact the only known champion machine that exhibits [[Counter]] behavior. It provides the lower bound:

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

== Cryptids ==
Known BB(2,5) Cryptids

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA}}, known as [[Hydra]].
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}, known as the bonus cryptid.

== Holdouts ==
In April 2024, Shawn Ligocki publicly released a list of 23,411 undecided BB(2,5) machines. Justin Blanchard then made substantial progress over the course of the next month, reducing the list to 499 holdouts in late May 2024. In June 2024, @mxdys cut down the list to 273 using halting and inductive deciders, and again to 217 using [[Closed Tape Language|CTL]]. This list of holdouts can be found [[:File:2x5 holdouts 217.txt|here]].

In February 2025, @mxdys ran a decider pipeline in [[Coq]] that resulted in only 173 holdouts. This list has some overlap with previous lists due to 20 machines previously solved by Justin Blanchard not yet being implemented in Coq. Therefore, the list of holdouts as of February 2025 is as low as 153.

[[Category:BB Domain]]

1RB3RB---1LB0LA 2LA4RA3LA4RB1LB

2025-02-26T23:36:03Z

Peacemaker II: Redirected page to Beaver Math Olympiad#Solved problems

#REDIRECT [[Beaver Math Olympiad#Solved problems]]

1RB1RB3LA4LA2RA 2LB3RA---3RA4RB

2025-02-26T23:35:34Z

Peacemaker II: Redirected page to Beaver Math Olympiad#Solved problems

#REDIRECT [[Beaver Math Olympiad#Solved problems]]

1RB0RB3LA4LA2RA 2LB3RA---3RA4RB

2025-02-26T23:34:08Z

Peacemaker II: Redirected page to Beaver Math Olympiad#Solved problems

#REDIRECT [[Beaver Math Olympiad#Solved problems]]

User talk:MrSolis

2025-02-17T17:20:23Z

Peacemaker II: /* Explaining Myself */

== Antihydra ==
Hi, I rolled back your recent change to [[Antihydra]] because it seems like you removed a ton of content there. Feel free to add your proofs if you'd like, but no need to remove the transition table and so much of the description. [[User:Sligocki|Sligocki]] ([[User talk:Sligocki|talk]]) 03:35, 17 February 2025 (UTC)
== Explaining Myself ==
[[User:Sligocki|Sligocki]]
Hello,

I appreciate your criticisms of the changes I have made to one of your pages. I would like to explain my thought process behind the changes so that we may come to an agreement on what should and should not be changed. I will review each section of your article to explain why I did what I did.

1. The lead section: I wanted to be brief. Perhaps I was being too brief.

2. Turing Machine: ''No'' other individual Turing Machine page has a transition ''table'', only its inline form. If you look at the pages for the 5-state busy beaver winner and Hydra, the other two TM pages that I have made significant changes to, you can infer that I intend for these to be the ''final'' (or semi-final, for unsolved machines) versions of these pages. That means I want consistency across every individual Turing Machine page and a clear structure. If you prefer that I not edit the table out, let me know so that I can add one to the 5-state busy beaver winner and Hydra (and any other TM pages I decide to modify) pages. I also omitted the names mxdys and Racheline in my version. Is that a problem? I did not think that was a problem because I had cited your blog post where they ''are'' mentioned. If it is a problem, I will not omit their names in my second edit to the page.

3. Analysis: You will notice that I did not use the preferred and more useful rules in my version. That is intentional. I noticed that the page "Hydra function" just says "The Hydra function is Collatz-like function whose behavior is connected to the the unsolved halting problems for the Cryptids Hydra and Antihydra". Aside from the obvious grammatical errors, this article does not explain ''how'' the function H(n) = floor((3*n)/2) is connected to Hydra and Antihydra except for the name, which feels circular. It also does not explain why that matters. After Antihydra, I was planning on changing that article so that these questions could be answered.

4. Biased Random Walk: I have two separate points on why I changed it.

The first and more important one, is that the probability analysis is simply '''incorrect.''' The probability of the random walker ''b'' going down from ''n'' to -1 is '''not''' (1/2)^(n+1). The simple argument for why is that (1/2)^(n+1) is the probability that the random coin lands on the same face n+1 times ''in a row''. Consider the situation where ''b'' = 5, and you are given a coin to flip. It if lands on heads, ''b'' increases by 2; if tails, decreases by 1. If you had to get ''b'' to -1 without getting heads a single time, then the probability ''would'' be (1/2)^6 (assuming independent random flips where heads and tails each have a 1/2 probability). However, if you just had to get b to -1 you could still win if you get heads on your first flip and then get eight tails in a row. You could win even if ''b'' rose to 10^(20) if you are lucky enough. These situations are not very likely to happen, but they make a non-zero difference.

My second point is that I did not entirely remove this section from the wiki; I had ''moved'' it to the Hydra page in the Trajectory section and corrected it, along with giving a proof for why my analysis is true there. I did not want to include basically the same thing in two separate pages, so in my Antihydra edit I basically just said "read the Hydra article".

If you still do not believe me when I say that (1/2)^(n+1) is incorrect, I would like you to make and run a program based on this pseudocode:

<pre>
trials = [some large integer]
fakeInfinity = [some other large integer]

integer reachedMinusOneCount = 0
integer startingValue = 1 // THIS IS N

for (integer i = 0; i < trials; i++)
integer walk = startingValue

while(walk > -1 AND walk < fakeInfinity)
integer flip = [either 0 or 1, chosen randomly]
if (flip equals 0)
walk = walk - 1 // Getting a 0 is tails here
else
walk = walk + 2 // Getting a 1 is heads here

if (walk equals -1)
increment reachedMinusOneCount

probability = (reachedMinusOneCount)/(trials)
output(probability)</pre>
If you alter startingValue and run the program, you will find that probability is greater than (1/2)^(n+1) and closer to what I got.

5. Simulation: I included the part where after 2^(31) rule steps we have b = 1073720884 in my Trajectory section. I thought that if the average uninformed reader was curious about the "20940 (0.002%)" error, they could check for themselves, so I removed that part. Again, the same problematic (1/2)^(n+1) probability shows up, which I saw no issue with removing.

I sincerely apologize for making drastic changes to your article and not elaborating on why before I did that. I would like for you to respond with what precisely I did wrong so that I do not make unwanted changes in the future.
[[User:MrSolis|MrSolis]] ([[User talk:MrSolis|talk]]) 15:30, 17 February 2025 (UTC)

: Thanks for your efforts in improving the wiki! Are you on the discord server? It might be easier to discuss wiki changes there since that's where most people are active. [[User:Peacemaker II|Peacemaker II]] ([[User talk:Peacemaker II|talk]]) 17:00, 17 February 2025 (UTC)

::[[User:Peacemaker II]], No, I am simply curious as to exactly where I went too far. [[User:MrSolis|MrSolis]] ([[User talk:MrSolis|talk]]) 17:05, 17 February 2025 (UTC)

::: Personally, I liked how the previous version of the article followed Shawn's blog post, which I think is a good explanation of the machine for most readers. Good job on writing down proofs, but I think the majority of readers will only be interested in the high level rules, not a detailed explanation of the Turing machine behavior. Adequately explaining the Turing machine behavior can be cumbersome and readers can get easily lost with the details. IMO, there is little gain in understanding the derivation rather than just knowing the high-level rules. Perhaps we can find another place for the detailed proofs? If you want to discuss further, it'll be easier to talk on the discord server: https://discord.com/invite/3uqtPJA9Uv [[User:Peacemaker II|Peacemaker II]] ([[User talk:Peacemaker II|talk]]) 17:19, 17 February 2025 (UTC)

User talk:MrSolis

2025-02-17T17:01:01Z

Peacemaker II: /* Explaining Myself */

BB(2,5)

2025-02-07T19:53:16Z

Peacemaker II: rewording

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

The current BB(2,5) champion is {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}, also discovered by Daniel Yuan in June 2024. It is a halting [[shift overflow counter]], and is in fact the only known champion machine that exhibits [[Counter]] behavior. It provides the lower bound:

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

== Cryptids ==
Known BB(2,5) Cryptids

* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB0LA}}, known as [[Hydra]].
* {{TM|1RB3RB---3LA1RA_2LA3RA4LB0LB1LB}}, known as the bonus cryptid.

== Holdouts ==
Justin Blanchard generated a list of 499 holdouts in May 2024, which was cut down to 273 by an application of deciders written for BB(6), then again to 217 using CTL by the end of June 2024. This list of holdouts can be found [[:File:2x5 holdouts 217.txt|here]]. Since then, between 50 and 100 machines have been claimed to have been solved, but no new reduced holdout list has been generated.

[[Category:BB Domain]]

BB(3,3)

2025-01-20T18:45:18Z

Peacemaker II:

The 3-state, 3-symbol Busy Beaver problem '''BB(3,3)''' is unsolved. With the discovery of the [[Cryptids|Cryptid]] machine [[Bigfoot]] in 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].

The current BB(3,3) champion {{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}} was discovered by [[User:tjligocki|Terry]] and [[User:sligocki|Shawn Ligocki]] in 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>

On [[bbchallenge.org]] Discord we have reduced the unofficial [[holdouts list]] to 22 TMs. One of these, {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}}, is under very active exploration in channel #bb3x3 and believed to be a [[probviously]] halting TM by some members. If it halts, it will be the new champion.

== Cryptids ==
Known BB(3,3) Cryptids:
* {{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}, known as [[Bigfoot]]
Possibly probviously halting Cryptid:

* {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}}

== Top Halters ==
The current top 10 BB(3,3) halters (known by [[User:sligocki|Shawn Ligocki]]) are
<pre>
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
</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 [https://discord.com/channels/960643023006490684/1259770474897080380/1325592156257124443 5 Jan 2025], @tligocki finished an enumeration and filtering of the BB(3,3) machines using the established Ligocki filters, listing the filter used for each machine. He also computed the number of steps and sigma scores for all found halting TMs. The thorough results are located [https://drive.google.com/drive/folders/101U9htddFdE889agAZnU58VLgQJlfn0v?usp=drive_link here]. 367 machines remained on that list.

Over two-thirds of the 367 remaining machines were shown to be non-halting with FAR and MITMWFAR by @Justin Blanchard on [https://discord.com/channels/960643023006490684/1259770474897080380/1262265485639286876 14 July 2024].

Most of the remainders were shown non-halting by @lijil on [https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 8 June 2023.]

These leaves 21 TMs, very similar to the more informal list identified below. The odd one out in the list below is {{TM|1RB2LA0LA_2LC---2RA_0RA2RC1LC|undecided}}, which is on @lijil's certified list.

== Holdouts ==
@Justin Blanchard's informal [[Holdouts lists|holdouts list]] is down to 22 TMs as of 9 Jun 2024. @Andrew Ducharme has [https://discord.com/channels/960643023006490684/1259770474897080380/1260476021212188763 categorized] them as follows:

Cryptids
* 829: Bigfoot: {{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}

Unsolved:
* {{TM|1RB1LB2LC_1LA2RB1RB_---0LA2LA|undecided}} (397)
* Group 2
** {{TM|1RB0LB0RC_2LC2LA1RA_1RA1LC---|undecided}} (153, equivalent to 758)
** {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}} (758, equivalent to 153)
* Group 3
** {{TM|1RB2LA1LA_2LA0RA2RC_---0LC2RA|undecided}} (531, equivalent to 532)
** {{TM|1RB2LA1LA_2LA0RA2RC_---1RB2RA|undecided}} (532, equivalent to 531)
Non-halting TMs with solutions/proofs of varying rigorousness:
* Group 1 ([https://discord.com/channels/960643023006490684/1259770474897080380/1291694606534049793 Discord summary of proofs])
** {{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|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)
** {{TM|1RB2RA1LB_0LC0RA1LA_2LA0RB---|undecided}} (818, equivalent to 21 and 92)
The halting problems for Group 1 machines are highly dependent on the halting problem for machine 816. If it is non-halting, then 21, 92, 683, 817, and 818 are all non-halting. If 816 halts via transition C0, then 817 halts. If 816 halts via transition C2, then 21, 92, 683 and 818 all halt.
* {{TM|1RB2LA1LC_1LA2RB1RB_---2LB0LC|undecided}} (543). See argument for non-halting by @dyuan on [https://discord.com/channels/960643023006490684/1084047886494470185/1240854539519791216 16 May 2025] and [https://discord.com/channels/960643023006490684/1259770474897080380/1325646208483590177 5 January 2025.]
* {{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|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|1RB0RC---_2RC0LB1LB_2LC2RA2RB|undecided}} (279). [https://discord.com/channels/960643023006490684/1084047886494470185/1256729514076016733 Longitudinal analysis by @Legion implies non-halting.]
* {{TM|1RB2LA2RA_1LC1LB0RA_2RA0LB---|undecided}} (637). Analyzed by [https://discord.com/channels/960643023006490684/1084047886494470185/1211655096098885732 @Justin], proven by [https://discord.com/channels/960643023006490684/1084047886494470185/1222028527755460700 @dyuan] and [https://discord.com/channels/960643023006490684/1259770474897080380/1327101773604720730 @Andrew].
* {{TM|1RB2RB---_1LC2LB1RC_0RA0LB1RB|undecided}} (834, related to Group 1). [https://discord.com/channels/960643023006490684/1084047886494470185/1222694817771814922 See argument] from @dyuan for similarity to 642.
* {{TM|1RB2RB1LC_1LA2RB0RB_2LB---0LA|undecided}} (867). [https://discord.com/channels/960643023006490684/1084047886494470185/1256634331464601640 Longitudinal analysis by @Legion implies non-halting.]

These TMs on the 9 Jun 2024 list have formal proofs.

* {{TM|1RB1LC---_0LC2RB1LB_2LA0RC1RC|undecided}} (400, equivalent to 494). [https://discord.com/channels/960643023006490684/1218877181321678928/1265131102289264690 Coq proof by @-d.] [https://discord.com/channels/960643023006490684/1084047886494470185/1215532512307052634 Equivalence argument provided by @Legion.]

* {{TM|1RB2LA0LA_2LC---2RA_0RA2RC1LC|undecided}} (494, equivalent to 400, non-halting certified on @lijil's list, equivalence not certified)
* {{TM|1RB2LB---_1RC2RB1LC_0LA0RB1LB|undecided}} (642, related to Group 1) ([https://cosearch.bbchallenge.org/contribution/3wxkcx6f cosearch]). Coq proof by @-d. [https://discord.com/channels/960643023006490684/1084047886494470185/1222694817771814922 See argument] for non-halting by @dyuan.

BB(2,5)

2025-01-20T16:14:48Z

Peacemaker II:

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

The current BB(2,5) champion is {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}, also discovered by Daniel Yuan in June 2024. It is notable for being the only champion machine that exhibits [[Counter]] behavior.

BB(2,5)

2025-01-20T16:13:39Z

Peacemaker II:

BB(3,3)

2025-01-20T16:10:21Z

Peacemaker II:

The 3-state, 3-symbol Busy Beaver problem '''BB(3,3)''' is unsolved. With the discovery of the [[Cryptids|Cryptid]] [[Bigfoot]] in 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].

The current BB(3,3) champion {{TM|0RB2LA1RA_1LA2RB1RC_1RZ1LB1LC|halt}} was discovered by [[User:tjligocki|Terry]] and [[User:sligocki|Shawn Ligocki]] in 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>

On [[bbchallenge.org]] Discord we have reduced the unofficial [[holdouts list]] to 22 TMs. One of these, {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}}, is under very active exploration in channel #bb3x3 and believed to be a [[probviously]] halting TM by some members. If it halts, it will be the new champion.

== Cryptids ==
Known BB(3,3) Cryptids:
* {{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}, known as [[Bigfoot]]
Possibly probviously halting Cryptid:

* {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}}

== Top Halters ==
The current top 10 BB(3,3) halters (known by [[User:sligocki|Shawn Ligocki]]) are
<pre>
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
</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 [https://discord.com/channels/960643023006490684/1259770474897080380/1325592156257124443 5 Jan 2025], @tligocki finished an enumeration and filtering of the BB(3,3) machines using the established Ligocki filters, listing the filter used for each machine. He also computed the number of steps and sigma scores for all found halting TMs. The thorough results are located [https://drive.google.com/drive/folders/101U9htddFdE889agAZnU58VLgQJlfn0v?usp=drive_link here]. 367 machines remained on that list.

Over two-thirds of the 367 remaining machines were shown to be non-halting with FAR and MITMWFAR by @Justin Blanchard on [https://discord.com/channels/960643023006490684/1259770474897080380/1262265485639286876 14 July 2024].

Most of the remainders were shown non-halting by @lijil on [https://discord.com/channels/960643023006490684/1084047886494470185/1116351783040716830 8 June 2023.]

These leaves 21 TMs, very similar to the more informal list identified below. The odd one out in the list below is {{TM|1RB2LA0LA_2LC---2RA_0RA2RC1LC|undecided}}, which is on @lijil's certified list.

== Holdouts ==
@Justin Blanchard's informal [[Holdouts lists|holdouts list]] is down to 22 TMs as of 9 Jun 2024. @Andrew Ducharme has [https://discord.com/channels/960643023006490684/1259770474897080380/1260476021212188763 categorized] them as follows:

Cryptids
* 829: Bigfoot: {{TM|1RB2RA1LC_2LC1RB2RB_---2LA1LA|undecided}}

Unsolved:
* {{TM|1RB1LB2LC_1LA2RB1RB_---0LA2LA|undecided}} (397)
* Group 2
** {{TM|1RB0LB0RC_2LC2LA1RA_1RA1LC---|undecided}} (153, equivalent to 758)
** {{TM|1RB2LC1RC_2LC---2RB_2LA0LB0RA|undecided}} (758, equivalent to 153)
* Group 3
** {{TM|1RB2LA1LA_2LA0RA2RC_---0LC2RA|undecided}} (531, equivalent to 532)
** {{TM|1RB2LA1LA_2LA0RA2RC_---1RB2RA|undecided}} (532, equivalent to 531)
Non-halting TMs with solutions/proofs of varying rigorousness:
* Group 1 ([https://discord.com/channels/960643023006490684/1259770474897080380/1291694606534049793 Discord summary of proofs])
** {{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|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)
** {{TM|1RB2RA1LB_0LC0RA1LA_2LA0RB---|undecided}} (818, equivalent to 21 and 92)
The halting problems for Group 1 machines are highly dependent on the halting problem for machine 816. If it is non-halting, then 21, 92, 683, 817, and 818 are all non-halting. If 816 halts via transition C0, then 817 halts. If 816 halts via transition C2, then 21, 92, 683 and 818 all halt.
* {{TM|1RB2LA1LC_1LA2RB1RB_---2LB0LC|undecided}} (543). See argument for non-halting by @dyuan on [https://discord.com/channels/960643023006490684/1084047886494470185/1240854539519791216 16 May 2025] and [https://discord.com/channels/960643023006490684/1259770474897080380/1325646208483590177 5 January 2025.]
* {{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|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|1RB0RC---_2RC0LB1LB_2LC2RA2RB|undecided}} (279). [https://discord.com/channels/960643023006490684/1084047886494470185/1256729514076016733 Longitudinal analysis by @Legion implies non-halting.]
* {{TM|1RB2LA2RA_1LC1LB0RA_2RA0LB---|undecided}} (637). Analyzed by [https://discord.com/channels/960643023006490684/1084047886494470185/1211655096098885732 @Justin], proven by [https://discord.com/channels/960643023006490684/1084047886494470185/1222028527755460700 @dyuan] and [https://discord.com/channels/960643023006490684/1259770474897080380/1327101773604720730 @Andrew].
* {{TM|1RB2RB---_1LC2LB1RC_0RA0LB1RB|undecided}} (834, related to Group 1). [https://discord.com/channels/960643023006490684/1084047886494470185/1222694817771814922 See argument] from @dyuan for similarity to 642.
* {{TM|1RB2RB1LC_1LA2RB0RB_2LB---0LA|undecided}} (867). [https://discord.com/channels/960643023006490684/1084047886494470185/1256634331464601640 Longitudinal analysis by @Legion implies non-halting.]

These TMs on the 9 Jun 2024 list have formal proofs.

* {{TM|1RB1LC---_0LC2RB1LB_2LA0RC1RC|undecided}} (400, equivalent to 494). [https://discord.com/channels/960643023006490684/1218877181321678928/1265131102289264690 Coq proof by @-d.] [https://discord.com/channels/960643023006490684/1084047886494470185/1215532512307052634 Equivalence argument provided by @Legion.]

* {{TM|1RB2LA0LA_2LC---2RA_0RA2RC1LC|undecided}} (494, equivalent to 400, non-halting certified on @lijil's list, equivalence not certified)
* {{TM|1RB2LB---_1RC2RB1LC_0LA0RB1LB|undecided}} (642, related to Group 1) ([https://cosearch.bbchallenge.org/contribution/3wxkcx6f cosearch]). Coq proof by @-d. [https://discord.com/channels/960643023006490684/1084047886494470185/1222694817771814922 See argument] for non-halting by @dyuan.

BB(2,5)

2025-01-20T16:09:36Z

Peacemaker II:

BB(2,5)

2025-01-20T16:04:59Z

Peacemaker II: April instead of May

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

The current BB(2,5) champion is {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}, also discovered by Daniel Yuan in June 2024.

BB(2,5)

2025-01-20T15:58:32Z

Peacemaker II:

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

The current BB(2,5) champion is {{TM|1RB3LA4RB0RB2LA_1LB2LA3LA1RA1RZ|halt}}, also discovered by Daniel Yuan in June 2024.

BB(2,5)

2025-01-20T15:47:45Z

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

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

Introduction to the Busy Beaver Function

2024-06-13T03:37:13Z

Peacemaker II:

The following is an informal explanation of what the Busy Beaver function means and how it is determined.

The values of the Busy Beaver function are a result of playing the **Busy Beaver game**.

Premise of the game:

Imagine you have a robot on an infinite 1-dimensional tape full of zeros. Your friend challenges you to create a program that will allow the robot to turn those zeros into as many ones as possible.

You think about it for a moment, and you have a genius idea. This is your program.

[State A]

If I see 0: Turn it into a 1, Move to the Right, Stay on State A

If I see 1: N/A

Your robot turns a zero into a one, and then moves to the right. Your robot will then see another zero. This will occur infinitely often, creating an infinite number of ones! You win, right?

Well, not quite. The busy beaver game forbids a program to have an infinite runtime (what use is a computer program that never ends?). In the busy beaver game, the robot must eventually halt.

Ok, you think. Here is a program that can create arbitrarily many ones and then halt:

[State A]

If I see 0: Turn it into a 1, Move to the Right, Go to State B

If I see 1: N/A

[State B]

If I see 0: Turn it into a 1, Move to the Right, Go to State C

If I see 1: N/A

[State C]

If I see 0: Turn it into a 1, Move to the Right, Go to State D

If I see 1: N/A

[State D]

If I see 0: Turn it into a 1, Move to the Right, Go to State E

If I see 1: N/A

[State E]

If I see 0: Turn it into a 1, Move to the Right, Go to State F

If I see 1: N/A

[State F]

If I see 0: Turn it into a 1, Move to the Right, Go to State HALT

If I see 1: N/A

A robot running this program will create 6 ones before halting. But you can add as many states as you like, and the program will create that many ones. So you've beaten the busy beaver game—you just need an extremely large number of states.

For this reason, the busy beaver game sets a limit on the number of states your program can use. If we set the limit to 2 states, you can make the following program. I strongly suggest you get a paper and pencil and run this program manually. This will allow you to get a feel for how busy beaver programs work. (Note: the robot will always start on a tape full of zeros and on state A)

[State A]

If I see 0: Turn it into a 1, Move to the Right, Go to State B

If I see 1: Keep it a 1, Move to the Left, Go to State B

[State B]

If I see 0: Turn it into a 1, Move to the Left, Go to State A

If I see 1: Keep it a 1, Move to the Right, Go to State HALT

If you ran this program, you will see that the robot reaches the halt state on step 6, and it creates 4 ones in the process. And it turns out that this program is the best you can do with 2 states. Therefore, the busy beaver value for 2 states is 4 ones and 6 steps. In other words:

BB_Σ(2) = 4

BB_S(2) = 6