BusyBeaverWiki - User contributions [en]

TMBR: May 2026

2026-10-07T03:42:33Z

Tjligocki: Modified the appearance of some link text from the raw link to text.

{{TMBRnav|April 2026|June 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).'' [[:Category:This Month in Beaver Research|This Month in Beaver Research]] for May 2026.

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

* Some new champions were found:
** BBµ(16) ≥ 47, found by Shawn on 5 May: <code>M(C(R(S, R(P(2,1), R(P(3,2), C(R(S, P(3,1)), P(5,3), P(5,2))))), P(1,1), S))</code>.[https://discord.com/channels/960643023006490684/1447627603698647303/1501347538287067267]
** BBµ(17) ≥ 2090, found by Shawn on 8 May: <code>M(C(R(P(1,1),R(P(2,1),R(R(R(S,C(S,P(3,2))),P(4,1)),P(5,2)))),P(1,1),S))</code>.[https://discord.com/channels/960643023006490684/1447627603698647303/1502314241766588468]
** BBµ(93) ≥ [[Graham's number]], designed by racheline on 19 May using a binary encoding of weak goodstein sequences.[https://discord.com/channels/960643023006490684/1447627603698647303/1506245738978480198][https://discord.com/channels/960643023006490684/1447627603698647303/1506295959049474068]
* Some new cryptids were hand-built:
** Size 56, by Shawn on 2 May (simulating 5x+1 problem starting at 7).[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Size 49, by aparker, star and Shawn on 3 May (simulating [[wikipedia:Brocard's_problem|Brocard's problem]]).[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf]
* The first non-trivial divergent GRF was found (size 15). It halts iff there exists some n ≥ 1 such that n+3 divides <math>Tetr(n) = \frac{n(n+1)(n+2)}{6}</math>.[https://discord.com/channels/960643023006490684/960643023530762341/1500584497542987776] aparker[https://discord.com/channels/960643023006490684/960643023530762341/1500587569514283098] and star[https://discord.com/channels/960643023006490684/960643023530762341/1500595210919346337] proved that there is no such n.
* On 5 May, Shawn implemented a size 133 GRF that non-trivially halts iff there is a counter-example to Fermat's Last Theorem: [https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/fermat.mgrf fermat.mgrf].[https://discord.com/channels/960643023006490684/1447627603698647303/1501235408913305663] Note that since Fermat's Last Theorem is true, this GRF does not halt; however, this GRF is still of interest due to the sheer complexity and length of the proof of the theorem.
* On 6 May, Shawn found a chaotic (potentially Cryptid) size 16 GRF: <code>M(C(R(S, R(C(S, P(2,1)), C(R(P(1,1), P(3,1)), P(4,2), P(4,1)))), P(1,1), S))</code>.[https://discord.com/channels/960643023006490684/1447627603698647303/1501444301245382707]
* All M(PRF) GRFs were decided up to size 12 with max score 5.[https://discord.com/channels/960643023006490684/1447627603698647303/1508151420258484354]
* All PRFs were decided up to size 22 with max score <math>\approx 2^{2^{230}}</math>. All except 11 were decided by direct simulation, but 11 overflowed u64 and needed to be evaluated by more clever programs.[https://discord.com/channels/960643023006490684/1447627603698647303/1508151420258484354]
[[SKI Calculus]]:
* On 7 May, Δ⁵ discovered some tetrational (size 15) and pentational (size 19) terms in the SKI calculus,[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=prev&oldid=7455] and some small champions for [[SKI Calculus#SK calculus|Ξ₀SK(n)]] for sizes 9 to 15.[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7455]
* By 7 May, an unknown person (probably tromp) translated 2014MELO03's lambda calculus champion to SKI calculus, proving the bound Ξ₀(25) > [[Graham's number]][https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7461] (falsely attributed to 2014MELO03).
* On 15 May, Δ⁵ proved the bound Ξ₀(22) > [[Graham's number]].[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7521]
* On 19 May, Δ⁵ proved Ξ₀(6) = 17 and Ξ₀SK(6) = 10.[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7543]
* On 22 May, Δ⁵ discovered smaller tetrational (size 13) and pentational (size 17) terms in the SKI calculus, and 50_ft_locker proved the bound Ξ₀(20) > [[Graham's number]].[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7583]
* On 23 May, Boone discovered new Ξ₀SK(n) champions for sizes 7 to 12[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7600] (and Ξ₀(n) champions for sizes 7 to 10[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7613]), as well as a Ξ₀(11) champion 2 days later.[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7614]
* On 25 May, Azerty introduced the Busy Beaver function for the BCKW system.[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7633]
* On 26 May, Δ⁵ discovered some tetrational (size 7) and Ackeramnn-growth (size 11) terms in the [[SKI Calculus#BCKW system|BCKW system]], and proved the bound Ξ₀BCKW(12) > [[Graham's number]].[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7639]
* On 29 May, Δ⁵ discovered an infinite family of terms in the SKI calculus with normal form growth rate fω⋅2(n).[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7700]
[[CounterScript]]:
* On 1 May, discord user sheep discovered a BBCS(65) champion beating [[Graham's number]].[https://discord.com/channels/960643023006490684/1484108791636033659/1499727568763617381] On 3 May, sheep improved this champion [https://discord.com/channels/960643023006490684/1484108791636033659/1509179336136327310 to size 64].[https://discord.com/channels/960643023006490684/1484108791636033659/1500384646922240084]
* On 6 May, sheep discovered a new champion for BBCS(21)[https://discord.com/channels/960643023006490684/1484108791636033659/1501545706601578527] and on 11 May a new champion for BBCS(25).[https://discord.com/channels/960643023006490684/1484108791636033659/1503294458861719574]
* On 13 May, sheep proved CounterScript Turing-complete via a translation from B*olfuck to Counterscript.[https://discord.com/channels/960643023006490684/1484108791636033659/1504113618231492628] 2 days later, she showed that CounterScript restricted to 3 registers was also Turing-complete.[https://discord.com/channels/960643023006490684/1484108791636033659/1504770973684203581]
[[Fractran]]:
* On 28 May, @-d began enumerating BBf(23).[https://discord.com/channels/960643023006490684/1438019511155691521/1509348792330358965] The enumeration was completed by 31 May, leaving an initial 790,335 holdouts, which where reduced to 29,250 holdouts.[https://discord.com/channels/960643023006490684/1438019511155691521/1510749997392724190]

TODO: Check for other BB-like functions' progress

== Misc ==

* mxdys made [https://github.com/ccz181078/busycoq/blob/BB6/verify/Inductive_inf.v his inductive decider] for [[History linear|history-linear]] machines compilable into an executable file.[https://discord.com/channels/960643023006490684/960655108578881597/1501260508051800318]
* TODO: BB Bignum Bakeoff

* GPU deciders: The original discussion of this starts with a [https://discord.com/channels/960643023006490684/960643023530762343/1507544052185301092 discord post], where David SG announced some work he was doing with GPU deciders. Subsequently, they shared a [https://github.com/David5768/David-BB7-GPU-Deciders Github repo], and contributed some new results for BB(6) TMs. TODO: List BB(6) TMs decided, other domains, additional details.

* LegionMammal shared exploration they have made into searching for alternative "Axiom of Infinity" like statements and which ones imply which others, which could eventually be useful for creating [[Logical independence|axiomatically undecidable]] machines.[https://discord.com/channels/960643023006490684/960643023530762341/1488636445148184729]

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(4,3)]]
|5,641,006
|5,127,263
|513,743
|9.11%
|-
|[[BB(3,4)]]
|12,049,358
|11,362,197
|687,161
|5.70%
|-
|[[BB(2,5)]]
|66
|65
|1
|1.52%
|-
|[[BB(2,6)]]
|536,112
|413,513
|122,599
|22.87%
|}

* [[BB(7)]]
** David SG begins directly simulating the 17.8 million BB(7) holdouts using one of their servers, simulating over 1 million holdouts to halting.[https://discord.com/channels/960643023006490684/1369339127652159509/1509784217515462747]

*[[BB(2,5)]]
**[https://discord.com/channels/960643023006490684/1259770421046411285/1510322250769760376 Andrew Ducharme found a FAR solution for a machine], which previously had an informal nonhalting argument. On June 1st, [https://discord.com/channels/960643023006490684/1259770421046411285/1510867717194911874 mxdys provided the Rocq proof], reducing the Rocq-holdout count to '''65'''.
*[[BB(4,3)]]
**Andrew Ducharme reduced the number of holdouts from 5,641,006 to '''5,127,263''', a '''9.11%''' reduction, with mxdys's new inductive decider.[https://discord.com/channels/960643023006490684/1084047886494470185/1502750172407533628]
*[[BB(3,4)]]
**XnoobSpeakable completed Stage 2 of [[BB(3,4)#Phase 3|Phase 3]], reducing to number of holdouts from 12,049,358 to '''11,362,197''', a '''5.70%''' reduction.
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 536,112 to '''527,232''' via Enumerate.py and TM-enum, a '''1.66%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1500218448951775383][https://discord.com/channels/960643023006490684/1084047886494470185/1501436678823477368]
**Using the new inductive decider by mxdys and TM-enum, Andrew Ducharme first reduced the number of holdouts from 527,232 to '''501,914'''[https://discord.com/channels/960643023006490684/1084047886494470185/1501664852953792513][https://discord.com/channels/960643023006490684/1084047886494470185/1501812428113973268], then to '''439,120''' again using the inductive decider.[https://discord.com/channels/960643023006490684/1084047886494470185/1502436063103291492] In total this was a '''16.71%''' reduction.
**Using FAR, Andrew Ducharme further reduced the number of holdouts from 439,120 to '''418,127''', a '''4.78%''' reduction.
**Using some mxdys deciders, Andrew Ducharme continued reducing the number of holdouts through May 17, from 418,127 to '''413,513''' (a '''1.10%''' reduction).[https://discord.com/channels/960643023006490684/1084047886494470185/1505277282648981504][https://discord.com/channels/960643023006490684/1084047886494470185/1506063539498913934][https://discord.com/channels/960643023006490684/1084047886494470185/1507614139563118694][https://discord.com/channels/960643023006490684/1084047886494470185/1508623369967370472]
*[[BB(2,7)]]
** Terry Ligocki enumerated 130K more subtasks, increasing the number of holdouts to '''1,087,732,936'''. A total of 350K subtasks out of the 1 million subtasks (or '''35%''') have been enumerated.[https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659]

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

TMBR: May 2026

2026-10-06T22:53:13Z

Tjligocki: /* Misc */ Fleshed out the GPU deciders TODO with some details and other possible TODOs that may not be necessary.

{{TMBRnav|April 2026|June 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).'' [[:Category:This Month in Beaver Research|This Month in Beaver Research]] for May 2026.

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

* Some new champions were found:
** BBµ(16) ≥ 47, found by Shawn on 5 May: <code>M(C(R(S, R(P(2,1), R(P(3,2), C(R(S, P(3,1)), P(5,3), P(5,2))))), P(1,1), S))</code>.[https://discord.com/channels/960643023006490684/1447627603698647303/1501347538287067267]
** BBµ(17) ≥ 2090, found by Shawn on 8 May: <code>M(C(R(P(1,1),R(P(2,1),R(R(R(S,C(S,P(3,2))),P(4,1)),P(5,2)))),P(1,1),S))</code>.[https://discord.com/channels/960643023006490684/1447627603698647303/1502314241766588468]
** BBµ(93) ≥ [[Graham's number]], designed by racheline on 19 May using a binary encoding of weak goodstein sequences.[https://discord.com/channels/960643023006490684/1447627603698647303/1506245738978480198][https://discord.com/channels/960643023006490684/1447627603698647303/1506295959049474068]
* Some new cryptids were hand-built:
** Size 56, by Shawn on 2 May (simulating 5x+1 problem starting at 7).[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/collatz.mgrf]
** Size 49, by aparker, star and Shawn on 3 May (simulating [[wikipedia:Brocard's_problem|Brocard's problem]]).[https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/brocard.mgrf]
* The first non-trivial divergent GRF was found (size 15). It halts iff there exists some n ≥ 1 such that n+3 divides <math>Tetr(n) = \frac{n(n+1)(n+2)}{6}</math>.[https://discord.com/channels/960643023006490684/960643023530762341/1500584497542987776] aparker[https://discord.com/channels/960643023006490684/960643023530762341/1500587569514283098] and star[https://discord.com/channels/960643023006490684/960643023530762341/1500595210919346337] proved that there is no such n.
* On 5 May, Shawn implemented a size 133 GRF that non-trivially halts iff there is a counter-example to Fermat's Last Theorem: [https://github.com/sligocki/etc/blob/main/gen_rec/mgrf/fermat.mgrf fermat.mgrf].[https://discord.com/channels/960643023006490684/1447627603698647303/1501235408913305663] Note that since Fermat's Last Theorem is true, this GRF does not halt; however, this GRF is still of interest due to the sheer complexity and length of the proof of the theorem.
* On 6 May, Shawn found a chaotic (potentially Cryptid) size 16 GRF: <code>M(C(R(S, R(C(S, P(2,1)), C(R(P(1,1), P(3,1)), P(4,2), P(4,1)))), P(1,1), S))</code>.[https://discord.com/channels/960643023006490684/1447627603698647303/1501444301245382707]
* All M(PRF) GRFs were decided up to size 12 with max score 5.[https://discord.com/channels/960643023006490684/1447627603698647303/1508151420258484354]
* All PRFs were decided up to size 22 with max score <math>\approx 2^{2^{230}}</math>. All except 11 were decided by direct simulation, but 11 overflowed u64 and needed to be evaluated by more clever programs.[https://discord.com/channels/960643023006490684/1447627603698647303/1508151420258484354]
[[SKI Calculus]]:
* On 7 May, Δ⁵ discovered some tetrational (size 15) and pentational (size 19) terms in the SKI calculus,[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=prev&oldid=7455] and some small champions for [[SKI Calculus#SK calculus|Ξ₀SK(n)]] for sizes 9 to 15.[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7455]
* By 7 May, an unknown person (probably tromp) translated 2014MELO03's lambda calculus champion to SKI calculus, proving the bound Ξ₀(25) > [[Graham's number]][https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7461] (falsely attributed to 2014MELO03).
* On 15 May, Δ⁵ proved the bound Ξ₀(22) > [[Graham's number]].[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7521]
* On 19 May, Δ⁵ proved Ξ₀(6) = 17 and Ξ₀SK(6) = 10.[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7543]
* On 22 May, Δ⁵ discovered smaller tetrational (size 13) and pentational (size 17) terms in the SKI calculus, and 50_ft_locker proved the bound Ξ₀(20) > [[Graham's number]].[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7583]
* On 23 May, Boone discovered new Ξ₀SK(n) champions for sizes 7 to 12[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7600] (and Ξ₀(n) champions for sizes 7 to 10[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7613]), as well as a Ξ₀(11) champion 2 days later.[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7614]
* On 25 May, Azerty introduced the Busy Beaver function for the BCKW system.[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7633]
* On 26 May, Δ⁵ discovered some tetrational (size 7) and Ackeramnn-growth (size 11) terms in the [[SKI Calculus#BCKW system|BCKW system]], and proved the bound Ξ₀BCKW(12) > [[Graham's number]].[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7639]
* On 29 May, Δ⁵ discovered an infinite family of terms in the SKI calculus with normal form growth rate fω⋅2(n).[https://wiki.bbchallenge.org/w/index.php?title=SKI_Calculus&diff=next&oldid=7700]
[[CounterScript]]:
* On 1 May, discord user sheep discovered a BBCS(65) champion beating [[Graham's number]].[https://discord.com/channels/960643023006490684/1484108791636033659/1499727568763617381] On 3 May, sheep improved this champion [https://discord.com/channels/960643023006490684/1484108791636033659/1509179336136327310 to size 64].[https://discord.com/channels/960643023006490684/1484108791636033659/1500384646922240084]
* On 6 May, sheep discovered a new champion for BBCS(21)[https://discord.com/channels/960643023006490684/1484108791636033659/1501545706601578527] and on 11 May a new champion for BBCS(25).[https://discord.com/channels/960643023006490684/1484108791636033659/1503294458861719574]
* On 13 May, sheep proved CounterScript Turing-complete via a translation from B*olfuck to Counterscript.[https://discord.com/channels/960643023006490684/1484108791636033659/1504113618231492628] 2 days later, she showed that CounterScript restricted to 3 registers was also Turing-complete.[https://discord.com/channels/960643023006490684/1484108791636033659/1504770973684203581]
[[Fractran]]:
* On 28 May, @-d began enumerating BBf(23).[https://discord.com/channels/960643023006490684/1438019511155691521/1509348792330358965] The enumeration was completed by 31 May, leaving an initial 790,335 holdouts, which where reduced to 29,250 holdouts.[https://discord.com/channels/960643023006490684/1438019511155691521/1510749997392724190]

TODO: Check for other BB-like functions' progress

== Misc ==

* mxdys made [https://github.com/ccz181078/busycoq/blob/BB6/verify/Inductive_inf.v his inductive decider] for [[History linear|history-linear]] machines compilable into an executable file.[https://discord.com/channels/960643023006490684/960655108578881597/1501260508051800318]
* TODO: BB Bignum Bakeoff

* GPU deciders: The original discussion of this starts with a discord post, https://discord.com/channels/960643023006490684/960643023530762343/1507544052185301092, where David SG announced some work he was doing with GPU deciders. Subsequently, they shared a Github repo, https://github.com/David5768/David-BB7-GPU-Deciders, and contributed some new results for BB(6) TMs. TODO: List BB(6) TMs decided, other domains, additional details.

* LegionMammal shared exploration they have made into searching for alternative "Axiom of Infinity" like statements and which ones imply which others, which could eventually be useful for creating [[Logical independence|axiomatically undecidable]] machines.[https://discord.com/channels/960643023006490684/960643023530762341/1488636445148184729]

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(4,3)]]
|5,641,006
|5,127,263
|513,743
|9.11%
|-
|[[BB(3,4)]]
|12,049,358
|11,362,197
|687,161
|5.70%
|-
|[[BB(2,5)]]
|66
|65
|1
|1.52%
|-
|[[BB(2,6)]]
|536,112
|413,513
|122,599
|22.87%
|}

* [[BB(7)]]
** David SG begins directly simulating the 17.8 million BB(7) holdouts using one of their servers, simulating over 1 million holdouts to halting.[https://discord.com/channels/960643023006490684/1369339127652159509/1509784217515462747]

*[[BB(2,5)]]
**[https://discord.com/channels/960643023006490684/1259770421046411285/1510322250769760376 Andrew Ducharme found a FAR solution for a machine], which previously had an informal nonhalting argument. On June 1st, [https://discord.com/channels/960643023006490684/1259770421046411285/1510867717194911874 mxdys provided the Rocq proof], reducing the Rocq-holdout count to '''65'''.
*[[BB(4,3)]]
**Andrew Ducharme reduced the number of holdouts from 5,641,006 to '''5,127,263''', a '''9.11%''' reduction, with mxdys's new inductive decider.[https://discord.com/channels/960643023006490684/1084047886494470185/1502750172407533628]
*[[BB(3,4)]]
**XnoobSpeakable completed Stage 2 of [[BB(3,4)#Phase 3|Phase 3]], reducing to number of holdouts from 12,049,358 to '''11,362,197''', a '''5.70%''' reduction.
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 536,112 to '''527,232''' via Enumerate.py and TM-enum, a '''1.66%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1500218448951775383][https://discord.com/channels/960643023006490684/1084047886494470185/1501436678823477368]
**Using the new inductive decider by mxdys and TM-enum, Andrew Ducharme first reduced the number of holdouts from 527,232 to '''501,914'''[https://discord.com/channels/960643023006490684/1084047886494470185/1501664852953792513][https://discord.com/channels/960643023006490684/1084047886494470185/1501812428113973268], then to '''439,120''' again using the inductive decider.[https://discord.com/channels/960643023006490684/1084047886494470185/1502436063103291492] In total this was a '''16.71%''' reduction.
**Using FAR, Andrew Ducharme further reduced the number of holdouts from 439,120 to '''418,127''', a '''4.78%''' reduction.
**Using some mxdys deciders, Andrew Ducharme continued reducing the number of holdouts through May 17, from 418,127 to '''413,513''' (a '''1.10%''' reduction).[https://discord.com/channels/960643023006490684/1084047886494470185/1505277282648981504][https://discord.com/channels/960643023006490684/1084047886494470185/1506063539498913934][https://discord.com/channels/960643023006490684/1084047886494470185/1507614139563118694][https://discord.com/channels/960643023006490684/1084047886494470185/1508623369967370472]
*[[BB(2,7)]]
** Terry Ligocki enumerated 130K more subtasks, increasing the number of holdouts to '''1,087,732,936'''. A total of 350K subtasks out of the 1 million subtasks (or '''35%''') have been enumerated.[https://discord.com/channels/960643023006490684/1084047886494470185/1492652604088516659]

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

TMBR: October 2026

2026-10-05T04:11:44Z

Tjligocki: /* Holdouts */ Fixed the total holdout count to match the raw data - mistakes were made in the incremental updates!

{{TMBRnav|September 2026|November 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).'' [[:Category:This Month in Beaver Research|This Month in Beaver Research]] for October 2026.

== BB Adjacent ==
[[Register machine|Register Machine]]:

* On 1 October, sheep discovered a 12-state, 2-register machine which halts iff 7 converges to 1 under the Collatz-like 5n+1 map, thus showing that MBB(12, 2) is hard.[https://discord.com/channels/960643023006490684/1552294888740098139/1555083694937604097]
* On the same day, @-d enumerated MBB(8), which left 7,891 holdouts after reduction, or 404 holdouts if an ai-created decider is used.[https://discord.com/channels/960643023006490684/1552294888740098139/1555102010473193513]
* On the same day, @mineralli discovered a family of champions for MBB(n, 1).[https://discord.com/channels/960643023006490684/1552294888740098139/1555268332465422508]
* On 2 October, @mineralli discovered a new champion for MBB(10, 2).[https://discord.com/channels/960643023006490684/1552294888740098139/1555502006372663337]
* On the same day, sheep discovered 2 new tetrational champions for MBB(10).[https://discord.com/channels/960643023006490684/1552294888740098139/1555512334846402573][https://discord.com/channels/960643023006490684/1552294888740098139/1555546324567335048]
* On the same day, Shawn Ligocki released an MBB(8) holdout list consisting of 145 holdouts,[https://discord.com/channels/960643023006490684/1552294888740098139/1555699448066342953] using another ai-created decider similar to CTL.[https://discord.com/channels/960643023006490684/1552294888740098139/1555699290285019147]
* On 3 October, Shawn Ligocki released another MBB(8) holdout list consisting of 46 holdouts,[https://discord.com/channels/960643023006490684/1552294888740098139/1555754019606106212] later reduced to 6[https://discord.com/channels/960643023006490684/1552294888740098139/1555772428997886004] (from 7[https://discord.com/channels/960643023006490684/1552294888740098139/1555759004444270633]) holdouts.
* On the same day, Shawn Ligocki discovered a cryptid amongst the 6 remaining holdouts, showing MBB(8) is hard.[https://discord.com/channels/960643023006490684/1552294888740098139/1555767330531967099]
* On the same day, sheep analyzed the rest of the holdouts, discovering 4 equivalences and another cryptid,[https://discord.com/channels/960643023006490684/1552294888740098139/1555897664078618675] meaning MBB(8) "probviously" = 3,394.
== Misc ==

* On 1 October, dyuan shared a subset of BB(6) holdouts, consisting of 24 machines which are expected to be easily analyzable by [[Longitudinal Analysis|longitudinal analysis]].[https://discord.com/channels/960643023006490684/1239205785913790465/1555244959840337973]

== Holdouts ==

* [[BB(2,5)]]
**On October 2, prurq used Aristotle AI to solve another one of the remaining holdouts, reducing the unofficial count to 38 out of initially 60 Rocq-proven.[https://discord.com/channels/960643023006490684/1259770421046411285/1555690509245087827]
*[[BB(2,7)]]
** Terry Ligocki enumerated 20K more subtasks, increasing the number of holdouts to '''3,125,433,611'''. All 1 million subtasks have been enumerated! This enumeration took from November 2025 to October 2026 using the resources I had available to work on this enumeration. Now we can start decreasing the number of holdouts!

[[Category:This Month in Beaver Research|2026-10]]

BB(2,7)

2026-10-05T04:10:40Z

Tjligocki: /* Stage 1 */ Corrected all the errors in the holdout counts to bring them in line with the actual data - doh!

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='10'|Terry Ligocki
|Yes
|309,868,847
|rowspan='10'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,370
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,206
|-
|50-59xxxx
|Yes
|315,047,998
|-
|60-69xxxx
|Yes
|309,911,121
|-
|70-79xxxx
|Yes
|315,355,885
|-
|80-89xxxx
|Yes
|309,900,123
|-
|90-99xxxx
|Yes
|315,121,515
|-
|'''Cumulative'''
| ---
| ---
|'''3,125,433,611'''
| ---
|-
|}

=== Stage 2 ===

This could be running the Ligocki's "Enumerate.py" Python code to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

BB(2,7)

2026-10-05T03:50:37Z

Tjligocki: /* Stage 2 */ Clarifying what Phase 1, Stage 2 might be.

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='10'|Terry Ligocki
|Yes
|309,868,865
|rowspan='10'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70-79xxxx
|Yes
|315,355,885
|-
|80-89xxxx
|Yes
|309,900,123
|-
|90-99xxxx
|Yes
|315,121,515
|-
|'''Cumulative'''
| ---
| ---
|'''3,125,388,042'''
| ---
|-
|}

=== Stage 2 ===

This could be running the Ligocki's "Enumerate.py" Python code to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: October 2026

2026-10-05T03:48:38Z

Tjligocki: /* Holdouts */ Now we decrease the holdouts!

{{TMBRnav|September 2026|November 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).'' [[:Category:This Month in Beaver Research|This Month in Beaver Research]] for October 2026.

== BB Adjacent ==
[[Register machine|Register Machine]]:

* On 1 October, sheep discovered a 12-state, 2-register machine which halts iff 7 converges to 1 under the Collatz-like 5n+1 map, thus showing that MBB(12, 2) is hard.[https://discord.com/channels/960643023006490684/1552294888740098139/1555083694937604097]
* On the same day, @-d enumerated MBB(8), which left 7,891 holdouts after reduction, or 404 holdouts if an ai-created decider is used.[https://discord.com/channels/960643023006490684/1552294888740098139/1555102010473193513]
* On the same day, @mineralli discovered a family of champions for MBB(n, 1).[https://discord.com/channels/960643023006490684/1552294888740098139/1555268332465422508]
* On 2 October, @mineralli discovered a new champion for MBB(10, 2).[https://discord.com/channels/960643023006490684/1552294888740098139/1555502006372663337]
* On the same day, sheep discovered 2 new tetrational champions for MBB(10).[https://discord.com/channels/960643023006490684/1552294888740098139/1555512334846402573][https://discord.com/channels/960643023006490684/1552294888740098139/1555546324567335048]
* On the same day, Shawn Ligocki released an MBB(8) holdout list consisting of 145 holdouts,[https://discord.com/channels/960643023006490684/1552294888740098139/1555699448066342953] using another ai-created decider similar to CTL.[https://discord.com/channels/960643023006490684/1552294888740098139/1555699290285019147]
* On 3 October, Shawn Ligocki released another MBB(8) holdout list consisting of 46 holdouts,[https://discord.com/channels/960643023006490684/1552294888740098139/1555754019606106212] later reduced to 6[https://discord.com/channels/960643023006490684/1552294888740098139/1555772428997886004] (from 7[https://discord.com/channels/960643023006490684/1552294888740098139/1555759004444270633]) holdouts.
* On the same day, Shawn Ligocki discovered a cryptid amongst the 6 remaining holdouts, showing MBB(8) is hard.[https://discord.com/channels/960643023006490684/1552294888740098139/1555767330531967099]
* On the same day, sheep analyzed the rest of the holdouts, discovering 4 equivalences and another cryptid,[https://discord.com/channels/960643023006490684/1552294888740098139/1555897664078618675] meaning MBB(8) "probviously" = 3,394.
== Misc ==

* On 1 October, dyuan shared a subset of BB(6) holdouts, consisting of 24 machines which are expected to be easily analyzable by [[Longitudinal Analysis|longitudinal analysis]].[https://discord.com/channels/960643023006490684/1239205785913790465/1555244959840337973]

== Holdouts ==

* [[BB(2,5)]]
**On October 2, prurq used Aristotle AI to solve another one of the remaining holdouts, reducing the unofficial count to 38 out of initially 60 Rocq-proven.[https://discord.com/channels/960643023006490684/1259770421046411285/1555690509245087827]
*[[BB(2,7)]]
** Terry Ligocki enumerated 20K more subtasks, increasing the number of holdouts to '''3,125,388,042'''. All 1 million subtasks have been enumerated! This enumeration took from November 2025 to October 2026 using the resources I had available to work on this enumeration. Now we can start decreasing the number of holdouts!

[[Category:This Month in Beaver Research|2026-10]]

TMBR: October 2026

2026-10-05T03:34:15Z

Tjligocki: /* Holdouts */ Completed the first enumeration (Phase 1, Stage 1) of BB(2,7)!

{{TMBRnav|September 2026|November 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).'' [[:Category:This Month in Beaver Research|This Month in Beaver Research]] for October 2026.

== BB Adjacent ==
[[Register machine|Register Machine]]:

* On 1 October, sheep discovered a 12-state, 2-register machine which halts iff 7 converges to 1 under the Collatz-like 5n+1 map, thus showing that MBB(12, 2) is hard.[https://discord.com/channels/960643023006490684/1552294888740098139/1555083694937604097]
* On the same day, @-d enumerated MBB(8), which left 7,891 holdouts after reduction, or 404 holdouts if an ai-created decider is used.[https://discord.com/channels/960643023006490684/1552294888740098139/1555102010473193513]
* On the same day, @mineralli discovered a family of champions for MBB(n, 1).[https://discord.com/channels/960643023006490684/1552294888740098139/1555268332465422508]
* On 2 October, @mineralli discovered a new champion for MBB(10, 2).[https://discord.com/channels/960643023006490684/1552294888740098139/1555502006372663337]
* On the same day, sheep discovered 2 new tetrational champions for MBB(10).[https://discord.com/channels/960643023006490684/1552294888740098139/1555512334846402573][https://discord.com/channels/960643023006490684/1552294888740098139/1555546324567335048]
* On the same day, Shawn Ligocki released an MBB(8) holdout list consisting of 145 holdouts,[https://discord.com/channels/960643023006490684/1552294888740098139/1555699448066342953] using another ai-created decider similar to CTL.[https://discord.com/channels/960643023006490684/1552294888740098139/1555699290285019147]
* On 3 October, Shawn Ligocki released another MBB(8) holdout list consisting of 46 holdouts,[https://discord.com/channels/960643023006490684/1552294888740098139/1555754019606106212] later reduced to 6[https://discord.com/channels/960643023006490684/1552294888740098139/1555772428997886004] (from 7[https://discord.com/channels/960643023006490684/1552294888740098139/1555759004444270633]) holdouts.
* On the same day, Shawn Ligocki discovered a cryptid amongst the 6 remaining holdouts, showing MBB(8) is hard.[https://discord.com/channels/960643023006490684/1552294888740098139/1555767330531967099]
* On the same day, sheep analyzed the rest of the holdouts, discovering 4 equivalences and another cryptid,[https://discord.com/channels/960643023006490684/1552294888740098139/1555897664078618675] meaning MBB(8) "probviously" = 3,394.
== Misc ==

* On 1 October, dyuan shared a subset of BB(6) holdouts, consisting of 24 machines which are expected to be easily analyzable by [[Longitudinal Analysis|longitudinal analysis]].[https://discord.com/channels/960643023006490684/1239205785913790465/1555244959840337973]

== Holdouts ==

* [[BB(2,5)]]
**On October 2, prurq used Aristotle AI to solve another one of the remaining holdouts, reducing the unofficial count to 38 out of initially 60 Rocq-proven.[https://discord.com/channels/960643023006490684/1259770421046411285/1555690509245087827]
*[[BB(2,7)]]
** Terry Ligocki enumerated 20K more subtasks, increasing the number of holdouts to '''3,125,388,042'''. All 1 million subtasks have been enumerated! This enumeration took from November 2025 to October 2026 using the resources I had available to work on this enumeration.

[[Category:This Month in Beaver Research|2026-10]]

BB(2,7)

2026-10-05T03:30:50Z

Tjligocki: /* Stage 1 */ Completed Phase 1, Stage 1 of the B(2,7) enumeration!

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='10'|Terry Ligocki
|Yes
|309,868,865
|rowspan='10'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70-79xxxx
|Yes
|315,355,885
|-
|80-89xxxx
|Yes
|309,900,123
|-
|90-99xxxx
|Yes
|315,121,515
|-
|'''Cumulative'''
| ---
| ---
|'''3,125,388,042'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: October 2026

2026-10-04T05:47:31Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration

{{TMBRnav|September 2026|November 2026}}

''This edition of TMBR is in progress and has not yet been released. Please add any notes you think may be relevant (including in the form a of a TODO with a link to any relevant Discord discussion).'' [[:Category:This Month in Beaver Research|This Month in Beaver Research]] for October 2026.

== BB Adjacent ==
[[Register machine|Register Machine]]:

* On 1 October, sheep discovered a 12-state, 2-register machine which halts iff 7 converges to 1 under the Collatz-like 5n+1 map, thus showing that MBB(12, 2) is hard.[https://discord.com/channels/960643023006490684/1552294888740098139/1555083694937604097]
* On the same day, @-d enumerated MBB(8), which left 7,891 holdouts after reduction, or 404 holdouts if an ai-created decider is used.[https://discord.com/channels/960643023006490684/1552294888740098139/1555102010473193513]
* On the same day, @mineralli discovered a family of champions for MBB(n, 1).[https://discord.com/channels/960643023006490684/1552294888740098139/1555268332465422508]
* On 2 October, @mineralli discovered a new champion for MBB(10, 2).[https://discord.com/channels/960643023006490684/1552294888740098139/1555502006372663337]
* On the same day, sheep discovered 2 new tetrational champions for MBB(10).[https://discord.com/channels/960643023006490684/1552294888740098139/1555512334846402573][https://discord.com/channels/960643023006490684/1552294888740098139/1555546324567335048]
* On the same day, Shawn Ligocki released an MBB(8) holdout list consisting of 145 holdouts,[https://discord.com/channels/960643023006490684/1552294888740098139/1555699448066342953] using another ai-created decider similar to CTL.[https://discord.com/channels/960643023006490684/1552294888740098139/1555699290285019147]
* On 3 October, Shawn Ligocki released another MBB(8) holdout list consisting of 46 holdouts,[https://discord.com/channels/960643023006490684/1552294888740098139/1555754019606106212] later reduced to 6[https://discord.com/channels/960643023006490684/1552294888740098139/1555772428997886004] (from 7[https://discord.com/channels/960643023006490684/1552294888740098139/1555759004444270633]) holdouts.
* On the same day, Shawn Ligocki discovered a cryptid amongst the 6 remaining holdouts, showing MBB(8) is hard.[https://discord.com/channels/960643023006490684/1552294888740098139/1555767330531967099]
* On the same day, sheep analyzed the rest of the holdouts, discovering 4 equivalences and another cryptid,[https://discord.com/channels/960643023006490684/1552294888740098139/1555897664078618675] meaning MBB(8) "probviously" = 3,394.
== Misc ==

* On 1 October, dyuan shared a subset of BB(6) holdouts, consisting of 24 machines which are expected to be easily analyzable by [[Longitudinal Analysis|longitudinal analysis]].[https://discord.com/channels/960643023006490684/1239205785913790465/1555244959840337973]

== Holdouts ==

* [[BB(2,5)]]
**On October 2, prurq used Aristotle AI to solve another one of the remaining holdouts, reducing the unofficial count to 38 out of initially 60 Rocq-proven.[https://discord.com/channels/960643023006490684/1259770421046411285/1555690509245087827]
*[[BB(2,7)]]
** Terry Ligocki enumerated 10K more subtasks, increasing the number of holdouts to '''3,090,258,485'''. A total of 990K subtasks out of the 1 million subtasks (or '''99%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-10]]

BB(2,7)

2026-10-04T05:44:54Z

Tjligocki: /* Stage 1 */ Added 99xxxx results

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='19'|Terry Ligocki
|Yes
|309,868,865
|rowspan='19'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70-79xxxx
|Yes
|315,355,885
|-
|80-89xxxx
|Yes
|309,900,123
|-
|90xxxx
|Yes
|35,179,655
|-
|91xxxx
|Yes
|27,935,612
|-
|92xxxx
|Yes
|35,097,239
|-
|93xxxx
|Yes
|27,901,630
|-
|94xxxx
|Yes
|35,046,877
|-
|95xxxx
|Yes
|27,965,087
|-
|96xxxx
|Yes
|35,094,471
|-
|97xxxx
|Yes
|27,940,281
|-
|98xxxx
|No
|TBD
|-
|99xxxx
|Yes
|27,831,106
|-
|'''Cumulative'''
| ---
| ---
|'''3,090,258,485'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: September 2026

2026-09-30T20:16:52Z

Tjligocki: /* Champions */ Update BB(2,7) enumeration

{{TMBRnav|August 2026|October 2026}}

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

== Blog Posts ==

* September 1, 2026. Nick Drozd. [https://nickdrozd.github.io/2026/09/01/tetrational-self-cleaning-turing-machine.html A Tetrational Self-Cleaning Turing Machine].

== Champions ==

* On September 1, nickdrozd discovered a new [[BLB|BLB(3,3)]] champion which runs for over 10↑↑6.54 steps before blanking the tape.[https://discord.com/channels/960643023006490684/1259770474897080380/1544448259437895680] Note that this is not a new [[BBB|BBB(3,3)]] champion.
* On September 9, @-d discovered a new [[BBf|BBf(24)]] champion which runs for over 9.000 × 1014,448 steps.[https://discord.com/channels/960643023006490684/1438019511155691521/1547182811713380383]

== BB Adjacent ==

* [[Cyclic Tag]]
** On September 1, Azerty enumerated all BBCT programs up to size 14, proving the values of BBCT(n) for n < 7.[https://discord.com/channels/960643023006490684/1542262029430292501/1544315541383155784]
** On September 2, sheep discovered new champions for BBCT(11),[https://discord.com/channels/960643023006490684/1542262029430292501/1544351459846529134] BBCT(13),[https://discord.com/channels/960643023006490684/1542262029430292501/1544368872973664337] and BBCT(14).[https://discord.com/channels/960643023006490684/1542262029430292501/1544633416064962611]
** On September 5, Azerty proved the value of BBCT(7).[https://discord.com/channels/960643023006490684/1542262029430292501/1545914357777571951]
TODO: Cellular Automata Busy Beaver (version 2)
* [[General Recursive Function]]
** On September 5, after Shawn Ligocki renewed interest in a certain chaotic BBµ(16) holdout last month, sheep devised a method to accelerate the simulation of the holdout.[https://discord.com/channels/960643023006490684/1447627603698647303/1545730338725171250] This allowed her to confirm that, if the holdout returned a value, it would be larger than 3.124 × 1016.[https://discord.com/channels/960643023006490684/1260977480316092527/1546531543290224701]
** On September 7, the aforementioned method was ported to Rust by @ZTS439, allowing them to extend the bound to 4.999 × 1017.[https://discord.com/channels/960643023006490684/1447627603698647303/1546666535911821413]

TODO: PLUSMINUS[https://discord.com/channels/960643023006490684/1243312334907375676/1546687220390567948]

* [[Fractran]]
** On September 13, @-d completed enumeration of BBf(24), leaving an initial 6,733,785 holdouts, which were reduced to 231,432 holdouts, or 10,838 holdouts if ai-created deciders are used.[https://discord.com/channels/960643023006490684/1438019511155691521/1548561426262790144]

TODO: [[Post Tag System]] (alternate definition)

* [[Register machine|Register Machine]]
** On September 18, @mineralli discovered a new MBB(9) champion.[https://discord.com/channels/960643023006490684/1243312334907375676/1550599643031670875] sheep then discovered four new champions for MBB(9) in the four following days.[https://discord.com/channels/960643023006490684/1243312334907375676/1551985291198210071]
** On September 25, sheep discovered an MBB(10) champion,[https://discord.com/channels/960643023006490684/1552294888740098139/1552360261900435537] which was shown to halt in around 102,629,437 steps by Shawn Ligocki.[https://discord.com/channels/960643023006490684/1552294888740098139/1552690832006447186] This champion prompted a rework of Ligocki's hydra-like rule accelerator.[https://discord.com/channels/960643023006490684/1552294888740098139/1553109975528902729]
** On September 29, @mineralli discovered a 5-register, 19-state machine which halts iff the Erdős ternary conjecture is false.

== Holdouts ==
* [[BB(6)]]
** By September 4, mxdys solved all 16 Counter Class 1 TMs.[https://discord.com/channels/960643023006490684/1239205785913790465/1545487019218698290]
** On September 4, Discord user fifi_49513 claimed to have proven a TM non-halting. [https://discord.com/channels/960643023006490684/1545543031468462150/1545543031468462150] This machine was later confirmed non-halting in Rocq on September 5 by mxdys. [https://discord.com/channels/960643023006490684/1545543031468462150/1545682001481175095]
** On September 8, mxdys released a new holdouts list of '''964''' machines up to equivalence.[https://discord.com/channels/960643023006490684/1239205785913790465/1546881471556620339]
** On September 14, an independent computation by Seiji extended the [[Space Needle]] trajectory to '''100,000,000''' terms: no power of 2 was ever encountered, and the most factors of 2 any term had was '''25''' (first occurring at n = 23,145,881; the previously documented state was >17,000,000 terms with max 24). The full final value (94,125,050 bits) is published with SHA-256 anchors, together with 400 checkpoints cross-verified by two independent implementations, and exact macro step-count laws of the machine (L0–L4; L1 and L3 formally proven). [https://github.com/oohzaim1-afk/space-needle-100M]
** On September 16, mxdys released another holdouts list of '''855''' machines up to equivalence.[https://discord.com/channels/960643023006490684/1239205785913790465/1549771931677364264]
** On September 24, mxdys released another holdouts list of '''815''' machines up to equivalence.[https://discord.com/channels/960643023006490684/1239205785913790465/1552681541401059350] The newly solved holdouts include:
*** [[BMO|BMO9]]
*** Two shift-overflow bell counters
*** 31 shift-overflow mixed-digits counters with similar behavior
*** two new translated cyclers (also shift-overflow mixed-digits counters) with high period
*** two other shift-overflow counters
**On September 24, Discord user q64 simulated a TM to halt[https://discord.com/channels/960643023006490684/1239205785913790465/1552399839269224498]
**The halting status of {{TM|1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---}} was formalized without new axioms by avoiding the use of Baker–Wüstholz theorem.
**On September 30, mxdys released another holdouts list of '''797''' machines up to equivalence.[https://discord.com/channels/960643023006490684/1239205785913790465/1554765260735512616]
*[[BB(2,5)]]
**On September 8, mxdys released a new holdouts list of '''60''' machines.[https://discord.com/channels/960643023006490684/1259770421046411285/1546880921318326282]
**On September 24, prurq used Aristotle AI and GPT6 to solve 20 of the 60 remaining holdouts.[https://discord.com/channels/960643023006490684/1259770421046411285/1552990521650319453]
* [[BB(2,7)]]
** Terry Ligocki enumerated 31K more subtasks, increasing the number of holdouts to '''3,062,427,379'''. A total of 980K subtasks out of the 1 million subtasks (or '''98%''') have been enumerated.

== Misc ==
* On September 2, RobinCodes began work on updating the BB(6) holdout [https://docs.google.com/spreadsheets/d/1mMp8bAcTFT91j7azn72liX8NSTwc2E_ozKnOGTfRCfw/edit?pli=1&gid=1330361301#gid=1330361301 spreadsheet].[https://discord.com/channels/960643023006490684/1239205785913790465/1544742905850236978]
* On September 5, [[User:XnoobSpeakable|XnoobSpeakable]] began an opt-in archival project of the bbchallenge Discord server.[https://discord.com/channels/960643023006490684/960643023530762341/1545742987710038088]
* On September 23, Joseph M. Shunia published a preprint presenting a 120-state TM that halts iff Riemann hypothesis is false.[https://arxiv.org/abs/2609.30306]
* Twitter user @_Mira___Mira_ claimed to have found a TKTK-state TM <TKTK update when final TM is published> whose halting status is [[Independence from ZFC|independent of ZFC]] with GPT-6 Astra.

BB(2,7)

2026-09-30T20:14:50Z

Tjligocki: /* Stage 1 */ Added 94xxxx, 96xxxx, 97xxxx

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='19'|Terry Ligocki
|Yes
|309,868,865
|rowspan='19'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70-79xxxx
|Yes
|315,355,885
|-
|80-89xxxx
|Yes
|309,900,123
|-
|90xxxx
|Yes
|35,179,655
|-
|91xxxx
|Yes
|27,935,612
|-
|92xxxx
|Yes
|35,097,239
|-
|93xxxx
|Yes
|27,901,630
|-
|94xxxx
|Yes
|35,046,877
|-
|95xxxx
|Yes
|27,965,087
|-
|96xxxx
|Yes
|35,094,471
|-
|97xxxx
|Yes
|27,940,281
|-
|98xxxx
|No
|TBD
|-
|99xxxx
|No
|TBD
|-
|'''Cumulative'''
| ---
| ---
|'''3,062,427,379'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: September 2026

2026-09-28T05:58:47Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration - 95% done!

{{TMBRnav|August 2026|October 2026}}

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

== BB Adjacent ==

* [[Cyclic Tag]]
** On September 1, Azerty enumerated all BBCT programs up to size 14, proving the values of BBCT(n) for n ≤ 7.[https://discord.com/channels/960643023006490684/1542262029430292501/1544315541383155784]
** On September 2, sheep discovered new champions for BBCT(11),[https://discord.com/channels/960643023006490684/1542262029430292501/1544351459846529134] BBCT(13),[https://discord.com/channels/960643023006490684/1542262029430292501/1544368872973664337] and BBCT(14).[https://discord.com/channels/960643023006490684/1542262029430292501/1544633416064962611]

* [[General Recursive Function]]
** On September 5, after Shawn Ligocki renewed interest in a certain chaotic BBµ(16) holdout last month, sheep devised a method to accelerate the simulation of the holdout.[https://discord.com/channels/960643023006490684/1447627603698647303/1545730338725171250] This allowed her to confirm that, if the holdout returned a value, it would be larger than 3.124 × 1016.[https://discord.com/channels/960643023006490684/1260977480316092527/1546531543290224701]
** On September 7, the aforementioned method was ported to Rust by @ZTS439, allowing them to extend the bound to 4.999 × 1017.[https://discord.com/channels/960643023006490684/1447627603698647303/1546666535911821413]

TODO: PLUSMINUS

* [[Fractran]]
** On September 9, @-d discovered a new BBf(24) champion which runs for over 9.000 × 1014,448 steps.[https://discord.com/channels/960643023006490684/1438019511155691521/1547182811713380383]
** On September 13, @-d completed enumeration of BBf(24), leaving an initial 6,733,785 holdouts, which were reduced to 231,432 holdouts, or 10,838 holdouts if ai-created deciders are used.[https://discord.com/channels/960643023006490684/1438019511155691521/1548561426262790144]

TODO: [[Post Tag System]] (alternate definition)

* [[Register machine|Register Machine]]
** On September 18, @mineralli discovered a new MBB(9) champion.[https://discord.com/channels/960643023006490684/1243312334907375676/1550599643031670875] sheep then discovered four new champions for MBB(9) in the four following days.[https://discord.com/channels/960643023006490684/1243312334907375676/1551985291198210071]

== Holdouts ==
* [[BB(6)]]
** By September 4, mxdys solved all 16 Counter Class 1 TMs.[https://discord.com/channels/960643023006490684/1239205785913790465/1545487019218698290]
** On September 4, Discord user fifi_49513 claimed to have proven a TM non-halting. [https://discord.com/channels/960643023006490684/1545543031468462150/1545543031468462150] This machine was later confirmed non-halting in Rocq on September 5 by mxdys. [https://discord.com/channels/960643023006490684/1545543031468462150/1545682001481175095]
** On September 8, mxdys released a new holdouts list of '''964''' machines up to equivalence.[https://discord.com/channels/960643023006490684/1239205785913790465/1546881471556620339]
** On September 14, an independent computation by Seiji extended the [[Space Needle]] trajectory to '''100,000,000''' terms: no power of 2 was ever encountered, and the most factors of 2 any term had was '''25''' (first occurring at n = 23,145,881; the previously documented state was >17,000,000 terms with max 24). The full final value (94,125,050 bits) is published with SHA-256 anchors, together with 400 checkpoints cross-verified by two independent implementations, and exact macro step-count laws of the machine (L0–L4; L1 and L3 formally proven). [https://github.com/oohzaim1-afk/space-needle-100M]
** On September 16, mxdys released another holdouts list of '''855''' machines up to equivalence.[https://discord.com/channels/960643023006490684/1239205785913790465/1549771931677364264]
** On September 24, mxdys released another holdouts list of '''815''' machines up to equivalence.[https://discord.com/channels/960643023006490684/1239205785913790465/1552681541401059350] The newly solved holdouts include:
*** [[BMO|BMO9]]
*** Two shift-overflow bell counters
*** 31 shift-overflow mixed-digits counters with similar behavior
*** two new translated cyclers (also shift-overflow mixed-digits counters) with high period
*** two other shift-overflow counters
**On September 24, Discord user q64 simulated a TM to halt[https://discord.com/channels/960643023006490684/1239205785913790465/1552399839269224498]
**The halting status of {{TM|1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB---}} was formalized without new axioms by avoiding the use of Baker–Wüstholz theorem.
* [[BB(2,7)]]
** Terry Ligocki enumerated 28K more subtasks, increasing the number of holdouts to '''2,964,345,750'''. A total of 950K subtasks out of the 1 million subtasks (or '''95%''') have been enumerated.

== Misc ==
* On September 5, [[User:XnoobSpeakable|XnoobSpeakable]] began an opt-in archival project of the bbchallenge Discord server.[[Category:This Month in Beaver Research|2026-09]][https://discord.com/channels/960643023006490684/960643023530762341/1545742987710038088]

BB(2,7)

2026-09-28T05:57:46Z

Tjligocki: /* Stage 1 */ Added 92xxxx, 93xxxx, and 95xxxx

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='17'|Terry Ligocki
|Yes
|309,868,865
|rowspan='17'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70-79xxxx
|Yes
|315,355,885
|-
|80-89xxxx
|Yes
|309,900,123
|-
|90xxxx
|Yes
|35,179,655
|-
|91xxxx
|Yes
|27,935,612
|-
|92xxxx
|Yes
|35,097,239
|-
|93xxxx
|Yes
|27,901,630
|-
|94xxxx
|No
|TBD
|-
|95xxxx
|Yes
|27,965,087
|-
|96xxxx
|No
|TBD
|-
|97xxxx
|No
|TBD
|-
|98-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,964,345,750'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: September 2026

2026-09-18T08:21:35Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration - 92% done!

{{TMBRnav|August 2026|October 2026}}

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

== BB Adjacent ==

TODO: [[Cyclic Tag]]

== Holdouts ==
* [[BB(6)]]
** By September 4, mxdys solved all 16 Counter Class 1 TMs.[https://discord.com/channels/960643023006490684/1239205785913790465/1545487019218698290]
** On September 4, Discord user fifi_49513 claimed to have proven a TM non-halting. [https://discord.com/channels/960643023006490684/1545543031468462150/1545543031468462150] This machine was later confirmed non-halting in Rocq on September 5 by mxdys. [https://discord.com/channels/960643023006490684/1545543031468462150/1545682001481175095]
** On September 14, an independent computation by Seiji extended the [[Space Needle]] trajectory to '''100,000,000''' terms: no power of 2 was ever encountered, and the most factors of 2 any term had was '''25''' (first occurring at n = 23,145,881; the previously documented state was >17,000,000 terms with max 24). The full final value (94,125,050 bits) is published with SHA-256 anchors, together with 400 checkpoints cross-verified by two independent implementations, and exact macro step-count laws of the machine (L0–L4; L1 and L3 formally proven). [https://github.com/oohzaim1-afk/space-needle-100M]
* [[BB(2,7)]]
** Terry Ligocki enumerated 25K more subtasks, increasing the number of holdouts to '''2,873,381,794'''. A total of 920K subtasks out of the 1 million subtasks (or '''92%''') have been enumerated.

== Misc ==
* On September 5, [[User:XnoobSpeakable|XnoobSpeakable]] began an opt-in archival project of the bbchallenge Discord server[[Category:This Month in Beaver Research|2026-09]] .[https://discord.com/channels/960643023006490684/960643023530762341/1545742987710038088]

BB(2,7)

2026-09-18T08:20:01Z

Tjligocki: /* Stage 1 */ Consolidated 80xxxx-89xxxx

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='14'|Terry Ligocki
|Yes
|309,868,865
|rowspan='14'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70-79xxxx
|Yes
|315,355,885
|-
|80-89xxxx
|Yes
|309,900,123
|-
|90xxxx
|Yes
|35,179,655
|-
|91xxxx
|Yes
|27,935,612
|-
|92xxxx
|No
|TBD
|-
|93xxxx
|No
|TBD
|-
|94xxxx
|No
|TBD
|-
|95-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,873,381,794'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

BB(2,7)

2026-09-18T08:17:41Z

Tjligocki: /* Stage 1 */ Added 87xxxx-91xxxx results

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='23'|Terry Ligocki
|Yes
|309,868,865
|rowspan='23'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70-79xxxx
|Yes
|315,355,885
|-
|80xxxx
|Yes
|26,686,275
|-
|81xxxx
|Yes
|35,313,171
|-
|82xxxx
|Yes
|26,627,542
|-
|83xxxx
|Yes
|35,401,092
|-
|84xxxx
|Yes
|26,666,095
|-
|85xxxx
|Yes
|35,277,325
|-
|86xxxx
|Yes
|26,670,023
|-
|87xxxx
|Yes
|35,299,731
|-
|88xxxx
|Yes
|26,673,155
|-
|89xxxx
|Yes
|35,285,714
|-
|90xxxx
|Yes
|35,179,655
|-
|91xxxx
|Yes
|27,935,612
|-
|92xxxx
|No
|TBD
|-
|93xxxx
|No
|TBD
|-
|94xxxx
|No
|TBD
|-
|95-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,873,381,794'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: September 2026

2026-09-08T20:57:55Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration

{{TMBRnav|August 2026|October 2026}}

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

== BB Adjacent ==

TODO: [[Cyclic Tag]]

== Holdouts ==
* [[BB(6)]]
** By September 4, mxdys solved all 16 Counter Class 1 TMs.[https://discord.com/channels/960643023006490684/1239205785913790465/1545487019218698290]
** On September 4, Discord user fifi_49513 claimed to have proven a TM non-halting. [https://discord.com/channels/960643023006490684/1545543031468462150/1545543031468462150] This machine was later confirmed non-halting in Rocq on September 5 by mxdys. [https://discord.com/channels/960643023006490684/1545543031468462150/1545682001481175095]
* [[BB(2,7)]]
** Terry Ligocki enumerated 20K more subtasks, increasing the number of holdouts to '''2,713,007,927'''. A total of 870K subtasks out of the 1 million subtasks (or '''87%''') have been enumerated.

== Misc ==
* On September 5, [[User:XnoobSpeakable|XnoobSpeakable]] began an opt-in archival project of the bbchallenge Discord server[[Category:This Month in Beaver Research|2026-09]] .[https://discord.com/channels/960643023006490684/960643023530762341/1545742987710038088]

BB(2,7)

2026-09-08T20:56:54Z

Tjligocki: /* Stage 1 */ Added 85xxxx results

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='18'|Terry Ligocki
|Yes
|309,868,865
|rowspan='18'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70-79xxxx
|Yes
|315,355,885
|-
|80xxxx
|Yes
|26,686,275
|-
|81xxxx
|Yes
|35,313,171
|-
|82xxxx
|Yes
|26,627,542
|-
|83xxxx
|Yes
|35,401,092
|-
|84xxxx
|Yes
|26,666,095
|-
|85xxxx
|Yes
|35,277,325
|-
|86xxxx
|Yes
|26,670,023
|-
|87xxxx
|No
|TBD
|-
|88xxxx
|No
|TBD
|-
|89xxxx
|No
|TBD
|-
|90-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,713,007,927'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

Main Page

2026-09-06T01:03:22Z

Tjligocki: Change the light orange to tan because if you choose dark mode in a browser then the white text was almost invisible on the light orange background. A tan backgroud seemed to work for both light mode and dark mode.

The [[Busy Beaver function]] BB (called ''S'' originally) was introduced by [[Tibor Radó]] in 1962 for 2-symbol [[Turing machines]] and later generalized to ''m''-symbol Turing machines:<ref>Radó, T. (1962), On Non-Computable Functions. Bell System Technical Journal, 41: 877-884. https://doi.org/10.1002/j.1538-7305.1962.tb00480.x</ref><ref>Brady, Allen H, and the Meaning of Life, 'The Busy Beaver Game and the Meaning of Life', in Rolf Herken (ed.), The Universal Turing Machine: A Half-Century Survey (Oxford, 1990; online edn, Oxford Academic, 31 Oct. 2023), https://doi.org/10.1093/oso/9780198537748.003.0009, accessed 8 June 2024.</ref>

{| class="wikitable"
|-
| BB(''n'', ''m'') = Maximum number of steps taken by a halting ''n''-state, ''m''-symbol Turing machine starting from a blank (all 0) tape
|}

The 2-symbol case BB(''n'', 2) is abbreviated as BB(''n''). The Busy Beaver function is not computable, but a few of its values are known:

{| class="wikitable"
|+ Small busy beaver values<ref>P. Michel, "[https://bbchallenge.org/~pascal.michel/ha.html Historical survey of Busy Beavers]".</ref>
! !!2-state!!3-state !!4-state!!5-state!!6-state
!7-state
|-
! 2-symbol
| [[BB(2)]] = 6
| [[BB(3)]] = 21
| [[BB(4)]] = 107
| [[BB(5)]] = 47,176,870
| style="background: orange;" | [[BB(6)]] > <math>2 \uparrow \uparrow \uparrow 5</math>
| style="background: orange;" | [[BB(7)]] > <math>2 \uparrow^{11} 2 \uparrow^{11} 3</math>
|-
! 3-symbol
| [[BB(2,3)]] = 38
| style="background: orange;" | [[BB(3,3)]] > <math>10^{17}</math>
| style="background: #ddc290;" | [[BB(4,3)]] > <math>10 \uparrow^{4} 4</math>
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
|-
! 4-symbol
| [[BB(2,4)]] = 3,932,964
| style="background: #ddc290;" | [[BB(3,4)]] > <math>2 \uparrow^{15} 5</math>
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
|-
! 5-symbol
| style="background: orange;" | [[BB(2,5)]] > <math>10\uparrow\uparrow 4</math>
| style="background: #ddc290;" | [[BB(3,5)]] > <math> f_\omega(2 \uparrow^{15} 5)</math>
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
|-
! 6-symbol
| style="background: #ddc290;" | [[BB(2,6)]] > <math>10 \uparrow\uparrow\uparrow 3</math>
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
| style="background: #ddc290;" |
|}

In the above table, cells are highlighted in orange when there are known [[Cryptids]] (mathematically-hard machines) in that class, and cells are highlighted in tan when the existence of a Cryptid is given by using a known one with fewer states or symbols.

1-state domains and 1-symbol domains are omitted in the table as [[BB(1,m)]] = 1 and [[BB(n,1)]] = n.

== About bbchallenge ==
[[bbchallenge]] is a massively collaborative research project whose general goal is to obtain more knowledge on the [[Busy Beaver function]]. In practice, it mainly consists in collaboratively building [[Deciders]], programs that automatically prove that some Turing machines do not halt. Other efforts also include:

* Formalising results using theorem provers (such as [https://en.wikipedia.org/wiki/Rocq Rocq])
* Maintaining [[Holdouts lists]] for small busy beaver values
* [[Analysis Tools and Techniques|Proving]] the behavior of [[:Category:Individual Machines|Individual machines]]
* Finding [[Cryptids]] (mathematically-hard machines)
* Searching for new [[Champions]]
* Building [[Accelerated Simulator|Accelerated simulator]]s to simulate halting machines faster
* Writing papers and giving talks about the Busy Beaver function, see [[Papers & Talks]]
* Exploring different [[:Category:Functions|Busy Beaver-style functions]] built from different systems of computation

In June 2024, bbchallenge achieved a significant milestone by proving in [https://en.wikipedia.org/wiki/Rocq Rocq] (previously known as Coq) that the 5th busy beaver value, [[BB(5)]], is equal to the lower bound found in 1989: 47,176,870.<ref>H. Marxen and J. Buntrock. Attacking the Busy Beaver 5.
Bulletin of the EATCS, 40, pages 247-251, February 1990. https://turbotm.de/~heiner/BB/mabu90.html</ref>

== This Month in Beaver Research (TMBR) ==
[[:Category:This Month in Beaver Research|This Month in Beaver Research]] (TMBR, pronounced "timber") is a monthly summary of Busy Beaver research progress. Here are the three most recent released entries:

* [[TMBR: April 2026]]
* [[TMBR: March 2026]]
* [[TMBR: February 2026]]

[[TMBR: May 2026]], [[TMBR: June 2026]], [[TMBR: July 2026]], [[TMBR: August 2026]], and [[TMBR: September 2026]] are currently works in progress.

This Year in Beaver Research (TYBR) is a yearly summary of Busy Beaver research progress. Its first edition, [[TYBR: 2025]] is currently a work in progress.

==Notes==
<references />

TMBR: September 2026

2026-09-06T00:49:40Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration

{{TMBRnav|August 2026|October 2026}}

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

== BB Adjacent ==

TODO: [[Cyclic Tag]]

== Holdouts ==
* [[BB(6)]]
** By September 4, mxdys solved all 16 Counter Class 1 TMs.[https://discord.com/channels/960643023006490684/1239205785913790465/1545487019218698290]
** On September 4, Discord user fifi_49513 claimed to have proven a TM non-halting. [https://discord.com/channels/960643023006490684/1545543031468462150/1545543031468462150] This machine was later confirmed non-halting in Rocq on September 5 by mxdys. [https://discord.com/channels/960643023006490684/1545543031468462150/1545682001481175095]
* [[BB(2,7)]]
** Terry Ligocki enumerated 10K more subtasks, increasing the number of holdouts to '''2,677,730,602'''. A total of 860K subtasks out of the 1 million subtasks (or '''86%''') have been enumerated.

== Misc ==
* On September 5, [[User:XnoobSpeakable|Xnoob545]] began an opt-in archival project of the bbchallenge Discord server.[https://discord.com/channels/960643023006490684/960643023530762341/1545742987710038088] [[Category:This Month in Beaver Research|2026-09]]

TMBR: August 2026

2026-09-06T00:48:29Z

Tjligocki: /* Holdouts */ Removed a few colons in the "Holdouts" section for consistency with previous months.

{{TMBRnav|July 2026|September 2026}}

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

== BB Adjacent ==

* [[TT|TT (Terminating Turmites)]]
** On August 3, Discord user creeperman7002 discovered a new step and score champion for TT(3,3), <code>1PB2PA0PC_2TB1PC2PB_2PB1PZ0PA</code>, running for 715,764,479 steps.
** On August 8, Discord user creeperman7002 discovered a new step champion for TT(3,3), <code>1PB0TC1PA_2PC1PZ2PA_2TC0PA2PC</code>, running for <math>2.27\cdot10^{248}</math> steps.

TODO: [[Pebble Automaton]]

TODO: [[Wang Tiles]]

TODO: [[SKI Calculus]]

TODO: [[Cyclic Tag]]

== Holdouts ==
* [[BB(6)]]
** On August 3, prurq found three machines to halt.[https://discord.com/channels/960643023006490684/1239205785913790465/1533572940459675791][https://discord.com/channels/960643023006490684/1239205785913790465/1533837709452251227][https://discord.com/channels/960643023006490684/1239205785913790465/1533889409332674763]
** On August 5, q64 [https://discord.com/channels/960643023006490684/1239205785913790465/1534370964350042263 simulated another TM] to halt.
** On August 15, Discord user jacobrschwartz [https://discord.com/channels/960643023006490684/960643023530762341/1538044554286993418 proved another TM] halting.
** Later that day, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1538109685033607198 released] a new holdouts list of '''1,016''' machines up to equivalence, this is a '''4.51%''' reduction compared to the previous holdouts lists.
** On August 24, Discord user flintt_6 informally [https://discord.com/channels/960643023006490684/1239205785913790465/1541418968806334515 proved another TM] halting.
** On August 26, q64 [https://discord.com/channels/960643023006490684/1239205785913790465/1542210565404098652 proved another TM] halting.
** On August 30, mxdys [https://discordapp.com/channels/960643023006490684/1239205785913790465/1543436534831906926 released] another holdouts list of '''1,003''' machines up to equivalence.
** On August 31, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1543878992195620884 released] another holdouts list of '''999''' machines up to equivalence.
*[[BB(2,7)]]
**Terry Ligocki enumerated 150K more subtasks, increasing the number of holdouts to '''2,651,060,579'''. A total of 850K subtasks out of the 1 million subtasks (or '''85%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-08]]

TMBR: July 2026

2026-09-06T00:43:55Z

Tjligocki: /* Holdouts */ Removed three colon to make the "Holdout" sections consistent with previous months.

{{TMBRnav|June 2026|August 2026}}

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

== General ==

* TODO: BB(5) Lean proof completed: https://discord.com/channels/960643023006490684/1218877181321678928/1530515342772666429
* TODO: BBB(4) proof progress: https://discord.com/channels/960643023006490684/1385498968011575366/1532076512927420556

== BB Adjacent ==
[[SKI Calculus]]:
* A new champion was discovered for Ξ₀(12).
* New champions were discovered for Ξ₀_SK(12), Ξ₀_SK(13), Ξ₀_SK(14), Ξ₀_SK(15), Ξ₀_SK(16) and Ξ₀_SK(17).
TODO: update

[[Lambda Calculus]]:
* [https://discord.com/channels/960643023006490684/1355653587824283678/1528855324881326181 BBλ(40) was solved] on 20 July, showing that BBλ(40) ≈ 10↑↑↑16. BBλ(40) is the first solved domain in any BB variant to reach pentational values.
TODO: Post Tag System

TODO: Wang tiles (https://discord.com/channels/960643023006490684/1529586687926210691)

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1,094
|1,064
|30
|2.74%
|-
|[[BB(4,3)]]
|4,784,443
|4,747,566
|36,877
|0.77%
|}
* [[BB(6)]]
** On 2 July, mxdys solved three holdouts in rocq.[https://discord.com/channels/960643023006490684/1239205785913790465/1522010493529751604][https://discord.com/channels/960643023006490684/1239205785913790465/1522209150699044904]
** On 3 July, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1522617640554860574 solved another holdout] in rocq using an LLM.
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1531681818439319604 released] a new holdouts list of '''1,064''' machines up to equivalence, this is a '''2.74%''' reduction compared to the previous holdouts list.
** On 28 July, prurq [https://discord.com/channels/960643023006490684/1239205785913790465/1531564686238027976 found a TM to halt]. This was [https://discord.com/channels/960643023006490684/1239205785913790465/1532416648533901503 verified by mxdys] in rocq two days later together with another machine.
** TODO: BMO 9 and BMO 10
* [[BB(4,3)]]
** Andrew Ducharme reduced the number of holdouts from 4,784,443 to '''4,747,566''', a '''0.77%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1522168741922607164][https://discord.com/channels/960643023006490684/1084047886494470185/1530496818155229294]
*[[BB(2,7)]]
**Terry Ligocki enumerated 180K more subtasks, increasing the number of holdouts to '''2,185,010,519'''. A total of 700K subtasks out of the 1 million subtasks (or '''70%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-07]]

TMBR: June 2026

2026-09-06T00:42:20Z

Tjligocki: /* Holdouts */ Removed a colon to make the "Holdout" sections all look the same.

{{TMBRnav|May 2026|July 2026}}

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

== Talks ==

* On 24 June, Tristan Stérin presented the [https://arxiv.org/abs/2509.12337 BB(5) paper] at STOC 2026 in Salt Lake City, USA.[https://discord.com/channels/960643023006490684/1151558585344593950/1519344431801827339]

== BB Adjacent ==
[[Lambda Calculus|Busy Beaver for Lambda Calculus]]:
* [https://discord.com/channels/960643023006490684/1355653587824283678/1514278149049811058 BBλ(39) was solved] on 10 June, showing that BBλ(39) <math>= 5\cdot{3^{3^{3^3}}} + 6</math>.
* Work began on BBλ(40), which was reduced to 2 holdouts[https://discord.com/channels/960643023006490684/1355653587824283678/1517706496606081127] and BBλ(41), which was reduced to 93 holdouts.
* A list of BBλ(42) holdouts was also added to the [https://docs.google.com/spreadsheets/d/1jZ6TK9m3xmXUlC69727T-8WwvhALcsp8FrK6DzgThtw/edit?gid=1544675322#gid=1544675322 BBλ-spreadsheet].[https://discord.com/channels/960643023006490684/1355653587824283678/1517130491512492083]
[[SKI Calculus]]:
* New champions were discovered for Ξ₀(10), Ξ₀(13), Ξ₀(14), Ξ₀(15), Ξ₀(16), Ξ₀(17), Ξ₀(18) and Ξ₀(19).
* New [[SKI Calculus#SK calculus|SK calculus]] champions were discovered for Ξ₀_SK(18) and Ξ₀_SK(26), showing Ξ₀_SK(26) to be larger than Graham's number.
* New [[SKI Calculus#BCKW system|BCKW system]] champions were discovered for Ξ₀_BCKW(7), Ξ₀_BCKW(8) and Ξ₀_BCKW(9).
[[CounterScript]]:
* On 12 June, sheep constructed a size 85 champion with score <math>2^{q(5)}</math>[https://discord.com/channels/960643023006490684/1484108791636033659/1514987825127358615], where q is a fast growing function arising from [https://en.wikipedia.org/wiki/Laver_table Laver tables]. A day later, sheep showed that BBCS(89 + k) <math>\geq 2^{P^{k}(0)}</math> where P(x) = q(x + 1).[https://discord.com/channels/960643023006490684/1484108791636033659/1515330464683004044]
* On 15 June, [[User:Sligocki|Shawn Ligocki]] discovered new champions for BBCS(13)[https://discord.com/channels/960643023006490684/1484108791636033659/1515955533641809950] and BBCS(14).[https://discord.com/channels/960643023006490684/1484108791636033659/1515954591445815459]
* On the same day, Azerty completed the enumeration of BBCS(12), leaving an initial 866 holdouts.[https://discord.com/channels/960643023006490684/1484108791636033659/1516066605912821791] A second enumeration the next day left only 477 holdouts.[https://discord.com/channels/960643023006490684/1484108791636033659/1516366973523591249] [TODO: bug found, may have to be removed]
* On June 20, sheep discovered two new champions ([https://discord.com/channels/960643023006490684/1484108791636033659/1517854233020338216 first], [https://discord.com/channels/960643023006490684/1484108791636033659/1517857681086611457 second]) for BBCS(19) in quick succession, as well as new champions for BBCS(24) and BBCS(25).[https://discord.com/channels/960643023006490684/1484108791636033659/1517888154722504985]
[[Fractran]]:
* On 1 June, [[User:Sligocki|Shawn Ligocki]] discovered a new BBf(23) champion which runs for over <math>4.393 \times 10^{124}</math> steps.[https://discord.com/channels/960643023006490684/1438019511155691521/1510769318978129970]
* BBf(23) holdouts were reduced by 27.20% from 29,250 to 21,295.
[[Register machine|Register Machines]]:
* @-d discovered new champions for MBB(8) and MBB(9).[https://discord.com/channels/960643023006490684/1243312334907375676/1514082985664712824]
* MBB(6) was decided to be 49.
[[General Recursive Function|General Recursive Functions]]:
* [[User:Sligocki|Shawn Ligocki]] enumerated all PRF's of size 23.[https://discord.com/channels/960643023006490684/1447627603698647303/1518981870254297138]
* On 24 June Shawn Ligocki discovered a new champion PRF of size 23.[https://discord.com/channels/960643023006490684/1447627603698647303/1519059600081555467]
* On 25 June Shawn Ligocki [https://discord.com/channels/960643023006490684/1447627603698647303/1519579483617755316 discovered a new BBµ(19) champion].
* On 26 June Shawn Ligocki discovered two new BBµ(17) champions in quick succession.[https://discord.com/channels/960643023006490684/1447627603698647303/1521270196193464521][https://discord.com/channels/960643023006490684/1447627603698647303/1521327240732737536]

[[Branching Beaver]]:

* Shawn Ligocki introduced the Branching Beaver (BrB(n,m)) function which is a variation of the Busy Beaver function, but where the tape is a graph where every node has 3 neighbors.
* Shawn computed exact values BrB(1) = 3 and BrB(2) = 10 and champions BrB(3) ≥ 52, BrB(4) ≥ 4636,[https://discord.com/channels/960643023006490684/1243312334907375676/1515457952104845485] BrB(2,3) ≥ 68, BrB(2,4) ≥ 332372. Enumeration sizes of BrB TMs is larger than for traditional BB TMs, so solving these values is expected to by harder than solving traditional BB.

== Misc ==
* On 19 June, Marcelo Fornet shared a Lean formalization proving the BB(4) value.[https://discord.com/channels/960643023006490684/1218877181321678928/1517441864226181120] They are working on extending it to BB(5).
* TODO: GPU deciders

== Holdouts ==
{| class="wikitable"
|+BB Holdout Reduction by Domain
!Domain
!Previous Holdout Count
!New Holdout Count
!Holdout Reduction
!% Reduction
|-
|[[BB(6)]]
|1,104
|1,094
|10
|0.91%
|-
|[[BB(7)]]
|17,823,260
|15,743,264
|2,079,996
|11.67%
|-
|[[BB(4,3)]]
|5,127,263
|4,784,443
|342,820
|6.69%
|-
|[[BB(2,6)]]
|413,513
|412,086
|1,427
|0.35%
|}
*[[BB(6)]]
**mxdys solved four TMs with FAR.[https://discord.com/channels/960643023006490684/1239205785913790465/1512100325673271407][https://discord.com/channels/960643023006490684/1239205785913790465/1512309665244123239][https://discord.com/channels/960643023006490684/1239205785913790465/1513904347422134374][https://discord.com/channels/960643023006490684/1239205785913790465/1516821341678866523]
**prurq solved another four TMs with FAR.[https://discord.com/channels/960643023006490684/1239205785913790465/1513898370606039080][https://discord.com/channels/960643023006490684/1239205785913790465/1515998682053345320][https://discord.com/channels/960643023006490684/1239205785913790465/1516787722537140345]
** A TM was [https://discord.com/channels/960643023006490684/1239205785913790465/1515209217106251827 shown to be non-halting] by hipparcos, which was later [https://discord.com/channels/960643023006490684/1239205785913790465/1516464297407021086 confirmed in Rocq] by mxdys.
** On 21 June, Peacemaker II [https://discord.com/channels/960643023006490684/1239205785913790465/1518173416870510652 solved another TM] by running FAR with high parameters.
** On 29 June, hipparcos [https://discord.com/channels/960643023006490684/1521145274142036069/1521145274142036069 showed another TM] to be non-halting. This was verified in rocq by mxdys the same day.[https://discord.com/channels/960643023006490684/1239205785913790465/1521199140220964954]
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1521203430046171146 released] a new holdouts list of '''1,094''' machines up to equivalence, this is a '''0.91%''' reduction compared to the previous holdouts list.
*[[BB(7)]]
**Using new GPU based deciders, David SG reduced the number of remaining holdouts from 17,823,260 to '''16,949,557''', a '''4.90%''' reduction.[https://discord.com/channels/960643023006490684/1369339127652159509/1512553945837605016]
**Andrew Ducharme further reduced the amount of holdouts by '''7.12%''' to '''15,743,264''' by running mxdys's FAR decider with various parameters.[https://discord.com/channels/960643023006490684/1369339127652159509/1516364912375627816]
*[[BB(4,3)]]
**Andrew Ducharme reduced the number of holdouts from 5,127,263 to '''4,784,443''', a '''6.69%''' reduction, using FAR.[https://discord.com/channels/960643023006490684/1084047886494470185/1518549639228424252][https://discord.com/channels/960643023006490684/1084047886494470185/1519630042643169360][https://discord.com/channels/960643023006490684/1084047886494470185/1520871028988055623]
*[[BB(2,6)]]
**Andrew Ducharme reduced the number of holdouts from 413,513 to '''412,086''', a '''0.35%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1512386039933698128]
*[[BB(2,7)]]
**Terry Ligocki enumerated 170K more subtasks, increasing the number of holdouts to '''1,623,113,079'''. A total of 520K subtasks out of the 1 million subtasks (or '''52%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-06]]

BB(2,7)

2026-09-06T00:38:12Z

Tjligocki: /* Stage 1 */ Added 86xxxx

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='17'|Terry Ligocki
|Yes
|309,868,865
|rowspan='17'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70-79xxxx
|Yes
|315,355,885
|-
|80xxxx
|Yes
|26,686,275
|-
|81xxxx
|Yes
|35,313,171
|-
|82xxxx
|Yes
|26,627,542
|-
|83xxxx
|Yes
|35,401,092
|-
|84xxxx
|Yes
|26,666,095
|-
|85xxxx
|No
|TBD
|-
|86xxxx
|Yes
|26,670,023
|-
|87xxxx
|No
|TBD
|-
|88xxxx
|No
|TBD
|-
|89-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,677,730,602'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: August 2026

2026-09-02T03:38:20Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration - 85% done!

{{TMBRnav|July 2026|September 2026}}

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

== BB Adjacent ==
TODO: [[TT]]

TODO: [[Pebble Automaton]]

TODO: [[Wang Tiles]]

TODO: [[SKI Calculus]]

== Holdouts ==
* [[BB(6)]]:
** prurq found three machines to halt.[https://discord.com/channels/960643023006490684/1239205785913790465/1533572940459675791][https://discord.com/channels/960643023006490684/1239205785913790465/1533837709452251227][https://discord.com/channels/960643023006490684/1239205785913790465/1533889409332674763]
** q64 [https://discord.com/channels/960643023006490684/1239205785913790465/1534370964350042263 simulated another TM] to halt.
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1538109685033607198 released] a new holdouts list of '''1,016''' machines up to equivalence, this is a '''4.51%''' reduction compared to the previous holdouts lists.
** Later, mxdys [https://discordapp.com/channels/960643023006490684/1239205785913790465/1543436534831906926 released] another holdouts list of '''1,003''' machines up to equivalence.
*[[BB(2,7)]]:
**Terry Ligocki enumerated 150K more subtasks, increasing the number of holdouts to '''2,651,060,579'''. A total of 850K subtasks out of the 1 million subtasks (or '''85%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-08]]

BB(2,7)

2026-09-02T03:37:15Z

Tjligocki: /* Stage 1 */ Added 83xxxx results - 85% done!

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='16'|Terry Ligocki
|Yes
|309,868,865
|rowspan='16'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70-79xxxx
|Yes
|315,355,885
|-
|80xxxx
|Yes
|26,686,275
|-
|81xxxx
|Yes
|35,313,171
|-
|82xxxx
|Yes
|26,627,542
|-
|83xxxx
|Yes
|35,401,092
|-
|84xxxx
|Yes
|26,666,095
|-
|85xxxx
|No
|TBD
|-
|86xxxx
|No
|TBD
|-
|87xxxx
|No
|TBD
|-
|88-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,651,060,579'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

BB(2,7)

2026-09-01T08:44:38Z

Tjligocki: /* Stage 1 */ Consolidated 70-79xxxx

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='15'|Terry Ligocki
|Yes
|309,868,865
|rowspan='15'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70-79xxxx
|Yes
|315,355,885
|-
|80xxxx
|Yes
|26,686,275
|-
|81xxxx
|Yes
|35,313,171
|-
|82xxxx
|Yes
|26,627,542
|-
|83xxxx
|No
|TBD
|-
|84xxxx
|Yes
|26,666,095
|-
|85xxxx
|No
|TBD
|-
|86xxxx
|No
|TBD
|-
|87-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,615,659,487'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: August 2026

2026-09-01T08:39:52Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration

{{TMBRnav|July 2026|September 2026}}

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

== BB Adjacent ==
TODO: [[TT]]

TODO: [[Pebble Automaton]]

TODO: [[Wang Tiles]]

TODO: [[SKI Calculus]]

== Holdouts ==
* [[BB(6)]]:
** prurq found three machines to halt.[https://discord.com/channels/960643023006490684/1239205785913790465/1533572940459675791][https://discord.com/channels/960643023006490684/1239205785913790465/1533837709452251227][https://discord.com/channels/960643023006490684/1239205785913790465/1533889409332674763]
** q64 [https://discord.com/channels/960643023006490684/1239205785913790465/1534370964350042263 simulated another TM] to halt.
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1538109685033607198 released] a new holdouts list of '''1,016''' machines up to equivalence, this is a '''4.51%''' reduction compared to the previous holdouts lists.
** Later, mxdys [https://discordapp.com/channels/960643023006490684/1239205785913790465/1543436534831906926 released] another holdouts list of '''1,003''' machines up to equivalence.
*[[BB(2,7)]]:
**Terry Ligocki enumerated 140K more subtasks, increasing the number of holdouts to '''2,615,659,487'''. A total of 840K subtasks out of the 1 million subtasks (or '''84%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-08]]

BB(2,7)

2026-09-01T08:39:09Z

Tjligocki: /* Stage 1 */ Added 79xxxx

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='24'|Terry Ligocki
|Yes
|309,868,865
|rowspan='24'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70xxxx
|Yes
|35,159,738
|-
|71xxxx
|Yes
|27,951,998
|-
|72xxxx
|Yes
|35,128,100
|-
|73xxxx
|Yes
|27,919,356
|-
|74xxxx
|Yes
|35,154,839
|-
|75xxxx
|Yes
|27,978,951
|-
|76xxxx
|Yes
|35,146,062
|-
|77xxxx
|Yes
|27,896,091
|-
|78xxxx
|Yes
|35,113,370
|-
|79xxxx
|Yes
|27,907,380
|-
|80xxxx
|Yes
|26,686,275
|-
|81xxxx
|Yes
|35,313,171
|-
|82xxxx
|Yes
|26,627,542
|-
|83xxxx
|No
|TBD
|-
|84xxxx
|Yes
|26,666,095
|-
|85xxxx
|No
|TBD
|-
|86xxxx
|No
|TBD
|-
|87-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,615,659,487'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: August 2026

2026-09-01T00:17:43Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration

{{TMBRnav|July 2026|September 2026}}

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

== BB Adjacent ==
TODO: [[TT]]

TODO: [[Pebble Automaton]]

TODO: [[Wang Tiles]]

TODO: [[SKI Calculus]]

== Holdouts ==
* [[BB(6)]]:
** prurq found three machines to halt.[https://discord.com/channels/960643023006490684/1239205785913790465/1533572940459675791][https://discord.com/channels/960643023006490684/1239205785913790465/1533837709452251227][https://discord.com/channels/960643023006490684/1239205785913790465/1533889409332674763]
** q64 [https://discord.com/channels/960643023006490684/1239205785913790465/1534370964350042263 simulated another TM] to halt.
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1538109685033607198 released] a new holdouts list of '''1,016''' machines up to equivalence, this is a '''4.51%''' reduction compared to the previous holdouts lists.
** Later, mxdys [https://discordapp.com/channels/960643023006490684/1239205785913790465/1543436534831906926 released] another holdouts list of '''1,003''' machines up to equivalence.
*[[BB(2,7)]]:
**Terry Ligocki enumerated 130K more subtasks, increasing the number of holdouts to '''2,587,752,107'''. A total of 830K subtasks out of the 1 million subtasks (or '''83%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-08]]

BB(2,7)

2026-09-01T00:16:11Z

Tjligocki: /* Stage 1 */ Added 81xxxx, 82xxxx, and 84xxxx

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='23'|Terry Ligocki
|Yes
|309,868,865
|rowspan='23'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70xxxx
|Yes
|35,159,738
|-
|71xxxx
|Yes
|27,951,998
|-
|72xxxx
|Yes
|35,128,100
|-
|73xxxx
|Yes
|27,919,356
|-
|74xxxx
|Yes
|35,154,839
|-
|75xxxx
|Yes
|27,978,951
|-
|76xxxx
|Yes
|35,146,062
|-
|77xxxx
|Yes
|27,896,091
|-
|78xxxx
|Yes
|35,113,370
|-
|79xxxx
|No
|TBD
|-
|80xxxx
|Yes
|26,686,275
|-
|81xxxx
|Yes
|35,313,171
|-
|82xxxx
|Yes
|26,627,542
|-
|83xxxx
|No
|TBD
|-
|84xxxx
|Yes
|26,666,095
|-
|85xxxx
|No
|TBD
|-
|86-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,587,752,107'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

BB(2,7)

2026-09-01T00:12:27Z

Tjligocki: /* Stage 1 */ Sneaking up on the final changes

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='23'|Terry Ligocki
|Yes
|309,868,865
|rowspan='23'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70xxxx
|Yes
|35,159,738
|-
|71xxxx
|Yes
|27,951,998
|-
|72xxxx
|Yes
|35,128,100
|-
|73xxxx
|Yes
|27,919,356
|-
|74xxxx
|Yes
|35,154,839
|-
|75xxxx
|Yes
|27,978,951
|-
|76xxxx
|Yes
|35,146,062
|-
|77xxxx
|Yes
|27,896,091
|-
|78xxxx
|Yes
|35,113,370
|-
|79xxxx
|No
|TBD
|-
|80xxxx
|Yes
|26,686,275
|-
|81xxxx
|No
|TBD
|-
|82xxxx
|No
|TBD
|-
|83xxxx
|No
|TBD
|-
|84xxxx
|No
|TBD
|-
|85xxxx
|No
|TBD
|-
|83xxxx
|No
|TBD
|-
|86-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,499,145,299'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

BB(2,7)

2026-09-01T00:10:37Z

Tjligocki: /* Stage 1 */ Trying to get changes to work

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='210'|Terry Ligocki
|Yes
|309,868,865
|rowspan='210'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70xxxx
|Yes
|35,159,738
|-
|71xxxx
|Yes
|27,951,998
|-
|72xxxx
|Yes
|35,128,100
|-
|73xxxx
|Yes
|27,919,356
|-
|74xxxx
|Yes
|35,154,839
|-
|75xxxx
|Yes
|27,978,951
|-
|76xxxx
|Yes
|35,146,062
|-
|77xxxx
|Yes
|27,896,091
|-
|78xxxx
|Yes
|35,113,370
|-
|79xxxx
|No
|TBD
|-
|80xxxx
|Yes
|26,686,275
|-
|81xxxx
|No
|TBD
|-
|82xxxx
|No
|TBD
|-
|83xxxx
|No
|TBD
|-
|84-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,499,145,299'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: August 2026

2026-08-24T20:20:30Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration

{{TMBRnav|July 2026|September 2026}}

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

== BB Adjacent ==
TODO: [[TT]]

TODO: [[Pebble Automaton]]

TODO: [[Wang Tiles]]

TODO: [[SKI Calculus]]

== Holdouts ==
* [[BB(6)]]:
** prurq found three machines to halt.[https://discord.com/channels/960643023006490684/1239205785913790465/1533572940459675791][https://discord.com/channels/960643023006490684/1239205785913790465/1533837709452251227][https://discord.com/channels/960643023006490684/1239205785913790465/1533889409332674763]
** q64 [https://discord.com/channels/960643023006490684/1239205785913790465/1534370964350042263 simulated another TM] to halt.
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1538109685033607198 released] a new holdouts list of '''1,016''' machines up to equivalence, this is a '''4.51%''' reduction compared to the previous holdouts list.
*[[BB(2,7)]]:
**Terry Ligocki enumerated 100K more subtasks, increasing the number of holdouts to '''2,499,145,299'''. A total of 800K subtasks out of the 1 million subtasks (or '''80%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-08]]

BB(2,7)

2026-08-24T20:19:20Z

Tjligocki: /* Stage 1 */ Added 78xxxx and 80xxxx results

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='21'|Terry Ligocki
|Yes
|309,868,865
|rowspan='21'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70xxxx
|Yes
|35,159,738
|-
|71xxxx
|Yes
|27,951,998
|-
|72xxxx
|Yes
|35,128,100
|-
|73xxxx
|Yes
|27,919,356
|-
|74xxxx
|Yes
|35,154,839
|-
|75xxxx
|Yes
|27,978,951
|-
|76xxxx
|Yes
|35,146,062
|-
|77xxxx
|Yes
|27,896,091
|-
|78xxxx
|Yes
|35,113,370
|-
|79xxxx
|No
|TBD
|-
|80xxxx
|Yes
|26,686,275
|-
|81xxxx
|No
|TBD
|-
|82xxxx
|No
|TBD
|-
|83xxxx
|No
|TBD
|-
|84-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,499,145,299'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: August 2026

2026-08-17T20:20:21Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration

{{TMBRnav|July 2026|September 2026}}

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

== BB Adjacent ==
TODO: [[TT]]

TODO: [[Pebble Automaton]]

TODO: [[Wang Tiles]]

TODO: [[SKI Calculus]]

== Holdouts ==
* [[BB(6)]]:
** prurq found three machines to halt.[https://discord.com/channels/960643023006490684/1239205785913790465/1533572940459675791][https://discord.com/channels/960643023006490684/1239205785913790465/1533837709452251227][https://discord.com/channels/960643023006490684/1239205785913790465/1533889409332674763]
** q64 [https://discord.com/channels/960643023006490684/1239205785913790465/1534370964350042263 simulated another TM] to halt.
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1538109685033607198 released] a new holdouts list of '''1,016''' machines up to equivalence, this is a '''4.51%''' reduction compared to the previous holdouts list.
*[[BB(2,7)]]:
**Terry Ligocki enumerated 80K more subtasks, increasing the number of holdouts to '''2,437,345,654'''. A total of 780K subtasks out of the 1 million subtasks (or '''78%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-08]]

BB(2,7)

2026-08-17T20:19:32Z

Tjligocki: /* Stage 1 */ Added 74xxxx-77xxxx results

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='18'|Terry Ligocki
|Yes
|309,868,865
|rowspan='18'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70xxxx
|Yes
|35,159,738
|-
|71xxxx
|Yes
|27,951,998
|-
|72xxxx
|Yes
|35,128,100
|-
|73xxxx
|Yes
|27,919,356
|-
|74xxxx
|Yes
|35,154,839
|-
|75xxxx
|Yes
|27,978,951
|-
|76xxxx
|Yes
|35,146,062
|-
|77xxxx
|Yes
|27,896,091
|-
|78xxxx
|No
|TBD
|-
|79xxxx
|No
|TBD
|-
|80xxxx
|No
|TBD
|-
|81-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,437,345,654'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: August 2026

2026-08-10T04:07:04Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration

{{TMBRnav|July 2026|September 2026}}

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

== BB Adjacent ==
TODO: [[TT]]

== Holdouts ==
* [[BB(6)]]:
** prurq found three machines to halt.[https://discord.com/channels/960643023006490684/1239205785913790465/1533572940459675791][https://discord.com/channels/960643023006490684/1239205785913790465/1533837709452251227][https://discord.com/channels/960643023006490684/1239205785913790465/1533889409332674763]
** q64 [https://discord.com/channels/960643023006490684/1239205785913790465/1534370964350042263 simulated another TM] to halt.
*[[BB(2,7)]]:
**Terry Ligocki enumerated 40K more subtasks, increasing the number of holdouts to '''2,311,169,711'''. A total of 740K subtasks out of the 1 million subtasks (or '''74%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-08]]

BB(2,7)

2026-08-10T04:06:10Z

Tjligocki: /* Stage 1 */ Added 72xxxx and 73xxxx results

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='14'|Terry Ligocki
|Yes
|309,868,865
|rowspan='14'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70xxxx
|Yes
|35,159,738
|-
|71xxxx
|Yes
|27,951,998
|-
|72xxxx
|Yes
|35,128,100
|-
|73xxxx
|Yes
|27,919,356
|-
|74xxxx
|No
|TBD
|-
|75xxxx
|No
|TBD
|-
|76xxxx
|No
|TBD
|-
|77.-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,311,169,711'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: August 2026

2026-08-06T06:51:28Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration

{{TMBRnav|July 2026|September 2026}}

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

== BB Adjacent ==
TODO: [[TT]]

== Holdouts ==
* [[BB(6)]]:
** prurq found three machines to halt.[https://discord.com/channels/960643023006490684/1239205785913790465/1533572940459675791][https://discord.com/channels/960643023006490684/1239205785913790465/1533837709452251227][https://discord.com/channels/960643023006490684/1239205785913790465/1533889409332674763]
** q64 [https://discord.com/channels/960643023006490684/1239205785913790465/1534370964350042263 simulated another TM] to halt.
*[[BB(2,7)]]:
**Terry Ligocki enumerated 20K more subtasks, increasing the number of holdouts to '''2,248,122,255'''. A total of 720K subtasks out of the 1 million subtasks (or '''72%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-08]]

BB(2,7)

2026-08-06T06:50:35Z

Tjligocki: /* Stage 1 */ Added 71xxxx results

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='12'|Terry Ligocki
|Yes
|309,868,865
|rowspan='12'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70xxxx
|Yes
|35,159,738
|-
|71xxxx
|Yes
|27,951,998
|-
|72xxxx
|No
|TBD
|-
|73xxxx
|No
|TBD
|-
|74xxxx
|No
|TBD
|-
|75-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,248,122,255'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: August 2026

2026-08-03T19:44:27Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration

{{TMBRnav|July 2026|September 2026}}

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

== BB Adjacent ==
TODO: [[TT]]

== Holdouts ==
* [[BB(6)]]:
** prurq found two machines to halt: one after <math>6.00\times 10^{1\,542}</math> steps, the other after <math>1.53\times 10^{699}</math> steps.[https://discord.com/channels/960643023006490684/1239205785913790465/1533572940459675791][https://discord.com/channels/960643023006490684/1239205785913790465/1533837709452251227]
*[[BB(2,7)]]:
**Terry Ligocki enumerated 10K more subtasks, increasing the number of holdouts to '''2,220,170,257'''. A total of 710K subtasks out of the 1 million subtasks (or '''71%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-08]]

BB(2,7)

2026-08-03T19:43:30Z

Tjligocki: /* Stage 1 */ Added 70xxxx results

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='11'|Terry Ligocki
|Yes
|309,868,865
|rowspan='11'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70xxxx
|Yes
|35,159,738
|-
|71xxxx
|No
|TBD
|-
|72xxxx
|No
|TBD
|-
|73xxxx
|No
|TBD
|-
|74-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,220,170,257'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: July 2026

2026-07-30T21:50:15Z

Tjligocki: /* Holdouts */ Update BB(2,7) enumeration - 70% done!

{{TMBRnav|June 2026|August 2026}}

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

== General ==

* TODO: BB(5) Lean proof completed: https://discord.com/channels/960643023006490684/1218877181321678928/1530515342772666429
* TODO: BBB(4) proof progress: https://discord.com/channels/960643023006490684/1385498968011575366/1532076512927420556

== BB Adjacent ==
[[SKI Calculus]]:
* A new champion was discovered for Ξ₀(12).
* New champions were discovered for Ξ₀_SK(12), Ξ₀_SK(13), Ξ₀_SK(14), Ξ₀_SK(15), Ξ₀_SK(16) and Ξ₀_SK(17).
TODO: update

[[Lambda Calculus]]:
* [https://discord.com/channels/960643023006490684/1355653587824283678/1528855324881326181 BBλ(40) was solved] on 20 July, showing that BBλ(40) ≈ 10↑↑↑16. BBλ(40) is the first solved domain in any BB variant to reach pentational values.
TODO: Post Tag System

== Holdouts ==
* [[BB(6)]]:
** On 2 July, mxdys solved three holdouts in rocq.[https://discord.com/channels/960643023006490684/1239205785913790465/1522010493529751604][https://discord.com/channels/960643023006490684/1239205785913790465/1522209150699044904]
** On 3 July, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1522617640554860574 solved another holdout] in rocq using an LLM.
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1531681818439319604 released] a new holdouts list of '''1,064''' machines up to equivalence, this is a '''2.74%''' reduction compared to the previous holdouts list.
* [[BB(4,3)]]:
** Andrew Ducharme reduced the number of holdouts from 4,784,443 to '''4,747,566''', a '''0.77%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1522168741922607164][https://discord.com/channels/960643023006490684/1084047886494470185/1530496818155229294]
*[[BB(2,7)]]:
**Terry Ligocki enumerated 180K more subtasks, increasing the number of holdouts to '''2,185,010,519'''. A total of 700K subtasks out of the 1 million subtasks (or '''70%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-07]]

BB(2,7)

2026-07-30T21:49:08Z

Tjligocki: /* Stage 1 */ Added 72xxxx row as this is being computed at the moment

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='10'|Terry Ligocki
|Yes
|309,868,865
|rowspan='10'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70xxxx
|No
|TBD
|-
|71xxxx
|No
|TBD
|-
|72xxxx
|No
|TBD
|-
|73-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,185,010,519'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

BB(2,7)

2026-07-30T21:47:43Z

Tjligocki: /* Stage 1 */ Added 67xxxx results and consolidated 60-69xxxx

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='9'|Terry Ligocki
|Yes
|309,868,865
|rowspan='9'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60-69xxxx
|Yes
|309,917,121
|-
|70xxxx
|No
|TBD
|-
|71xxxx
|No
|TBD
|-
|72-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,185,010,519'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

TMBR: July 2026

2026-07-30T09:28:33Z

Tjligocki: /* BB Adjacent */ Update BB(2,7) enumeration

{{TMBRnav|June 2026|August 2026}}

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

== General ==

* TODO: BB(5) Lean proof completed: https://discord.com/channels/960643023006490684/1218877181321678928/1530515342772666429
* TODO: BBB(4) proof progress: https://discord.com/channels/960643023006490684/1385498968011575366/1532076512927420556

== BB Adjacent ==
[[SKI Calculus]]:
* A new champion was discovered for Ξ₀(12).
* New champions were discovered for Ξ₀_SK(12), Ξ₀_SK(13), Ξ₀_SK(14), Ξ₀_SK(15), Ξ₀_SK(16) and Ξ₀_SK(17).
TODO: update

[[Lambda Calculus]]:
* [https://discord.com/channels/960643023006490684/1355653587824283678/1528855324881326181 BBλ(40) was solved] on 20 July, showing that BBλ(40) ≈ 10↑↑↑16. BBλ(40) is the first solved domain in any BB variant to reach pentational values.
TODO: Post Tag System

== Holdouts ==
* [[BB(6)]]:
** On 2 July, mxdys solved three holdouts in rocq.[https://discord.com/channels/960643023006490684/1239205785913790465/1522010493529751604][https://discord.com/channels/960643023006490684/1239205785913790465/1522209150699044904]
** On 3 July, mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1522617640554860574 solved another holdout] in rocq using an LLM.
** mxdys [https://discord.com/channels/960643023006490684/1239205785913790465/1531681818439319604 released] a new holdouts list of '''1,064''' machines up to equivalence, this is a '''2.74%''' reduction compared to the previous holdouts list.
* [[BB(4,3)]]:
** Andrew Ducharme reduced the number of holdouts from 4,784,443 to '''4,747,566''', a '''0.77%''' reduction.[https://discord.com/channels/960643023006490684/1084047886494470185/1522168741922607164][https://discord.com/channels/960643023006490684/1084047886494470185/1530496818155229294]
*[[BB(2,7)]]:
**Terry Ligocki enumerated 170K more subtasks, increasing the number of holdouts to '''2,149,787,578'''. A total of 690K subtasks out of the 1 million subtasks (or '''69%''') have been enumerated.

[[Category:This Month in Beaver Research|2026-07]]

BB(2,7)

2026-07-30T09:25:50Z

Tjligocki: /* Stage 1 */ Added 68xxxx and 69xxxx results

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='18'|Terry Ligocki
|Yes
|309,868,865
|rowspan='18'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60xxxx
|Yes
|26,700,472
|-
|61xxxx
|Yes
|35,320,029
|-
|62xxxx
|Yes
|26,700,270
|-
|63xxxx
|Yes
|35,329,410
|-
|64xxxx
|Yes
|26,669,417
|-
|65xxxx
|Yes
|35,348,829
|-
|66xxxx
|Yes
|26,657,839
|-
|67xxxx
|No
|TBD
|-
|68xxxx
|Yes
|26,710,203
|-
|69xxxx
|Yes
|35,257,711
|-
|70xxxx
|No
|TBD
|-
|71xxxx
|No
|TBD
|-
|72-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,149,787,578'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]

BB(2,7)

2026-07-30T09:22:53Z

Tjligocki: /* Stage 1 */ Added 66xxxx results

The 2-state, 7-symbol Busy Beaver problem, '''BB(2,7)''', 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(2,7). The current BB(2,7) [[champion]] is simply inherited from [[BB(2,6)]] as BB(2,7) is just starting to be explored.

== Top Halters ==

These are the 20 highest scoring machines that were found in the current investigation of BB(2,7). This list does not include TMs from smaller domains:

== Phase 1 ==

The initial phase of enumeration and reduction of [[holdouts]] started in November 2025 using [https://github.com/ccz181078/TM C++ code] mxdys shared. This work is ongoing and further described below. After this is completed, it is planned to run the Ligockis' '''Enumerate.py''' code to further reduce the number of holdout TMs.

=== Stage 1 ===

The code provided by mxdys breaks up the BB(2,7) enumeration into 1 million subtasks which each run for ~10 minutes and leave ~2500 holdouts based on an average of the first 1K subtasks. These values are about 5 times longer than and 25 times larger than the ones for [[BB(7)]].

This enumeration began in November 2025 and is summarized in the table below. The mxdys code '''main.cpp''' from [https://github.com/ccz181078/TM C++ code] was used with two modification: '''#define BB7''' was replaced with '''#define BB27''' and the following block of code was added in the section with BB domain definitions:
<pre>
#ifdef BB27
constexpr int64_t N_STATE=2;
constexpr int64_t N_CHAR=7;
constexpr int64_t log2_N_STATE=1;
constexpr int64_t log2_N_CHAR=3;
typedef uint8_t chr_t;
#endif
</pre>

Results:

{| class="wikitable sortable" style="text-align: center"
!Task range
!Done by
!Completed
!# holdouts
!Source
|-
|00-09xxxx
|rowspan='17'|Terry Ligocki
|Yes
|309,868,865
|rowspan='17'|[https://drive.google.com/drive/folders/11AiZYiKJq7v0ns9o5nt-xUsSgSpcuNvZ?usp=drive_link Google Drive]
|-
|10-19xxxx
|Yes
|315,243,072
|-
|20-29xxxx
|Yes
|309,968,380
|-
|30-39xxxx
|Yes
|315,040,474
|-
|40-49xxxx
|Yes
|309,976,212
|-
|50-59xxxx
|Yes
|314,996,395
|-
|60xxxx
|Yes
|26,700,472
|-
|61xxxx
|Yes
|35,320,029
|-
|62xxxx
|Yes
|26,700,270
|-
|63xxxx
|Yes
|35,329,410
|-
|64xxxx
|Yes
|26,669,417
|-
|65xxxx
|Yes
|35,348,829
|-
|66xxxx
|Yes
|26,657,839
|-
|67xxxx
|No
|TBD
|-
|68xxxx
|No
|TBD
|-
|69xxxx
|No
|TBD
|-
|70xxxx
|No
|TBD
|-
|71-99xxxx
|TBD
|No
|TBD
| ---
|-
|'''Cumulative'''
| ---
| ---
|'''2,087,819,664'''
| ---
|-
|}

=== Stage 2 ===

The Ligocki's "Enumerate.py" Python code is used to further reduce the number of holdouts and explore more of the [[TNF]] of BB(2,7).

[[Category:BB Domains]]