TMBR: August 2026

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BBB(4) = 32,779,478 was proven this month by @carrino using Claude AI. The first 71 steps of the champion, 1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch), are depicted in this spacetime diagram, using Shawn Ligocki's Visual Simulator. (Credit: Shawn Ligocki)

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

This Month in Beaver Research for August 2026.

This month saw us finally cross the long-awaited milestone of less than 1,000 holdouts in BB(6), after a flurry of simulations, as well as proofs from users both new and old, resulting in 65 holdouts being solved this month. mxdys has also added the 9th and 10th problems to the Beaver Math Olympiad. BB(4,3) recorded small but substantial reductions in holdouts, while Terry Ligocki's enumeration of BB(2,7) slowly but surely reaches the 85% mark.

This month was also quite a busy one for BB-adjacent functions. Most notably, for Beeping Busy Beaver functions, BBB(4) = 32,779,478, proven by @carrino! BBλ(41) = 5 × 333 + 6 has also been proven this month, and BBf(23) has been almost completely solved, save for 12 Cryptids and 1 oddball machine. Speaking of Fractran, @-d has begun the enumeration BBf(24), with Shawn Ligocki finding a machine that demonstrates BBf(24) > 9.263 × 109,595. creeperman7002 has been working in the field of Terminating Turmites, finding a machine that demonstrates TT(3,3) > 2.27 × 10248, while reducing the domain to under 20,000 holdouts. Δ⁵ has also been incrementally finding new large champions for the SKI and SK calculi, and the BCKW system, notably finding 6 new hexational-level champions in the calculi!

The Busy Beaver functions for Pebble Automata and Cyclic Tag have been introduced; Collatz-like behavior has already been detected amongst small pebble automata, but no Cryptids have been found as of yet, unlike in cyclic tag, where a size 36 Cryptid has been found. Diophantine Equations have also received some research this month through attempts to find Cryptids and new champions, many of which are related to the sums of three cubes problem. A new family of champions has also been found for the Busy Beaver function for Wang Tiles, and Shawn Ligocki has recently revived discussion about a small but chaotic and possible Cryptid General Recursive Function. A primality test and a program that computes the Fibonacci numbers have also been created in CounterScript.

There has also been extensive discussion on what should classify a machine or a program as a Cryptid, due to the wide range of potential Cryptids in BB(6) (most notably BMO1) and how virtually all of the known machines or programs classified as Cryptids rely, at least in part, on Collatz-like randomness; a new defintion of Cryptid may even include solved machines with behaviors reminiscent of diabolically hard but solved problems, such as Fermat's Last Theorem.

An archive of the Discord server is also being considered, for accessibility purposes and to prevent sources from being lost; however, they may be some ethical concerns and problems with the suggestion as it stands currently.

General

  • On August 2, @carrino, using Claude AI solved the last 290 holdouts remaining in the Coq proof, meaning BBB(4) is solved![1] Specifically, @carrino proved that BBB(4) = 32,779,478, the champion being 1RB1LC_1RD1RB_0RD0RC_1LD1LA (bbch), discovered by Nick Drozd in July 2021, which quasihalts by turning into a period 1 translated cycler after 32,779,478 steps.[2]

BB Adjacent

  • Fractran
    • On August 1, @-d used Opus 4.8 to write two deciders, potentially reducing the number of BBf(23) holdouts from 21,233 to 694.[3] (Note that the new holdout lists are unofficial, as the correctness of the deciders has not yet been verified).
    • On the same day, @-d used Opus 4.8 to decide 602 of the 694 remaining holdouts individually, and to create equivalences and implications between the remaining holdouts. As a result, 21 holdouts remain in the unofficial holdout list for BBf(23).[4]
    • On the same day, Shawn Ligocki isolated one of the 21 remaining holdouts (#601 in the original 694 holdout list) as non-Hydra-like. They say that its behavior is a mix of Collatz-like and BMO1.[5]
    • On August 2, @-d noticed that holdout #601 iterates a chaotic map, likely meaning it has no stable orbits that would otherwise allow for an easy proof.[6]
    • On August 3, @-d identified 19 of the 21 remaining size 23 holdouts as Hydra-like; the two remaining holdouts (#601 and #163) were described as "ugly" and "messy" respectively.[7]
    • On August 4, @-d used Opus 4.8 to prove holdout #163 non-halting, as it fell into a stable orbit; 4 holdouts have been removed due to equivalence, and another 3 have been removed due to implications regarding other holdouts. 12 of the remaining 13 holdouts follow different Hydra-like mechanisms and are Cryptids (a total of 90 size 23 machines are equivalent to one of these 12 holdouts), with the remaining holdout (#601) having been previously described as an ugly version of BMO1.[8]
    • On August 22, @-d enumerated 20% of BBf(24) (the complete ennumeration is expected to take 6 times longer than that of BBf(23).[9]), discovering a new champion which runs for over 4.804 × 101,184 steps.[10]
    • On the same day, Shawn Ligocki discovered another new BBf(24) champion which runs for over 9.263 × 109,595 steps.[11]
    • On August 26, Shawn Ligocki discovered two new size 24 halters (non-champions) which have Collatz-like and BMO1-like behaviors.[12]
  • Terminating Turmites
    • On August 2, Discord user creeperman7002 discovered a new step and score champion for TT(3,3), 1PB2PA0PC_2TB1PC2PB_2PB1PZ0PA, running for 715,764,479 steps.[13]
    • On August 8, Discord user creeperman7002 discovered a new step champion for TT(3,3), 1PB0TC1PA_2PC1PZ2PA_2TC0PA2PC, running for over 2.27 × 10248 steps.[14]
    • On August 11, Discord user creeperman7002 released a TT(3,3) holdouts list consisting of 19,368 holdouts.[15]
  • SKI Calculus
    • On August 6, Δ⁵ discovered a new champion for Ξ₀SK(16).[16]
    • On the same day, Δ⁵ discovered a new champion for Ξ₀BCKW(6).[17]
    • On August 14, Δ⁵ discovered another new champion for Ξ₀SK(16).[18]
    • On August 20, Δ⁵ discovered new hexational champions for Ξ₀(17), Ξ₀(18), and Ξ₀(19) in the SKI calculus, and Ξ₀SK(23), Ξ₀SK(24), and Ξ₀SK(25) in the SK calculus.[19]
    • On the same day, Δ⁵ discovered another new champion for Ξ₀SK(25).[20]
  • Lambda Calculus
    • On August 7, BBλ(41) was solved![21] We now know that BBλ(41) = 5 × 333 + 6 after solving 237 holdouts, thanks to the help of 2014MELO03, cppdecksucker, mxdys, and others.[22]
      • As a result, the bet between Sam and cppdecksucker (which began in April[23]) has also concluded; cppdecksucker, who lost the bet on whether there would be any Cryptids shorter than 42 bits, has donated 50 NZD to the Mermaids charity.[24]
AlephSquirrel has found a new champion for peBBle(9), which runs for 63,577 steps, by modifying mathmasterzach's 323 step 9-pebble automaton. The first 200 steps of this champion are displayed here using AlephSquirrel's simulator, with colors corresponding to the numbers of pebbles in a single cell. (Credit: AlephSquirrel)
  • Pebble Automaton
    • On August 8, AlephSquirrel introduced the Busy Beaver function for Pebble Automata, peBBle(n), proving the values of peBBle(n) for sizes 1 to 3.[25]
    • On the same day, sheep discovered an infinite family of automata, including the champion for peBBle(4), showing peBBle(n) ≥ n(n−1)2. [26]
    • On August 9, AlephSquirrel discovered a new champion for peBBle(5).[27]
    • On August 10, Deccy discovered a new champion for peBBle(6)[28]
    • On August 11, Deccy discovered a new champion for peBBle(7).[29]
    • On August 13, mathmasterzach discovered 3 new peBBle(8) champions in quick succession,[30][31][32] and the previous peBBle(9) champion, which runs for 323 steps.[33]
    • On August 16, AlephSquirrel proved the peBBle(4) champion to be optimal.[34]
    • On August 17, AlephSquirrel tweaked the peBBle(9) champion, creating a new automaton which runs for 63,577 steps, a new bound for peBBle(9).[35]
  • Diophantine Equation
    • On August 12, Shawn Ligocki discovered new bounds for BBdio(n) for heights 40, 48, 57, and 66, based on work done in the sums of three cubes problem.[36][37]
    • On the same day, Shawn Ligocki suggested calling the equation x3+y3+z3=114, and similar equations from the sums of three cubes problem, Cryptids,[38][39] though their classification was later changed to that of energy vampire,[40] following an objection by sheep;[41][42] she suggested calling them "energy vampired [sic]",[43] as these equations are expected to be solvable in practice, though they may be computationally hard to find solutions for.[44]
    • On August 14, Shawn Ligocki rediscovered papers (and a post on StackExchange) from Bogdan Grechuk that imply BBdio(29) > 106,[45] and that BBdio(31) is decidable.[46]
    • On August 23, star discovered the height 78 equation a4+b4−c4+30=0, which is expected to not have any solutions, but may be very difficult to prove.[47]. She also discovered the height 135 equation w5+x5+y5+z5−7=0,[48] which may be difficult to prove unsatisfiable, in much the same way that the sums of three cubes problem is hard.
  • CounterScript
    • On August 13, Azerty made a size 16 program which returns the nth Fibonacci number.[49]
    • On August 15, sheep made a size 35 program that tests the primality of a number.[50]
    • On August 21, Shawn Ligocki completed the simulation of BBCS(14) out to 10,000 steps (using their own way of counting steps), finding no machines that improve the existing bound of BBCS(14) ≥ 129.[51]
    • On August 24, sheep discovered a set of Collatz-like programs,[52] some of which simulate random unbiased walks.[53][54][55] However, they are not champions, due to the existence of smaller tetrational and pentational machines.
    • On the same day, Shawn Ligocki released a personal BBCS(12) holdouts list consisting of 13,580 holdouts, used for direct simulation.[56]
A new champion for BBWT(9) can be obtained from a generalization of sheep's BBWT(7) champion, found this month. This new champion is a set of 9 tiles, which can tile any n×n square up to n = 17, and no larger; in this image of the maximal 17×17 square patch, created using sheep's tile viewer, tiling rules are denoted by colors, similar to other depictions of Wang tiles. (Credit: sheep)
  • Wang Tiles
    • On August 14, sheep discovered a new BBWT(7) champion.[57]
    • On the same day, Azerty made a statement implying the lower bound BBWT(n) ≥ 3n - 10.[58]
  • Cyclic Tag
    • On August 26, Azerty introduced the Busy Beaver function for Cyclic Tag Systems, BBCT(n),[59] proving the values of BBCT(n) for sizes 1 to 4, and providing lower bounds for BBCT(n) for sizes 1 to 12.[60]
    • On the same day, sheep rediscovered literature that provided lower bounds for BBCT(n) for sizes 1 to 9.[61]
    • On August 30, sheep discovered a size 36 Cryptid, showing BBCT(36) is hard.[62]
  • General Recursive Function
    • On August 27, Shawn Ligocki renewed interest in a size 16 chaotic (possibly Cryptid) GRF previously discovered in May,[63] during a discussion on what behaviors should be required to classify a Cryptid.[64]

Holdouts

BB Holdout Reduction by Domain
Domain New Holdout Count Previous Holdout Count Holdout Reduction % Reduction
BB(6) 999 1,064 65 6.11%
BB(4,3) 4,720,332 4,747,566 27,234 0.57%
prurq simulated an impressive 4 BB(6) holdouts to halting this month; one of the machines simulated, 1RB0LB_0RC1RF_1LD1RE_0LA0RB_0RD0LD_0LE--- (bbch), is shown here in mxdys's general accelerated simulator using longitudinal analysis, which prurq used to confirm the result. (Image source: prurq)
  • BB(6)
    • On August 1, mxdys added BMO9 (1RB1LA_1RC0RD_1LA---_1RE1RD_1LF0LA_---0LE (bbch)) and BMO10 (1RB0LB_0RC0LD_1LD1RE_0LA0RF_0RD1RF_0RC--- (bbch)) to the Beaver Math Olympiad.[65][66]
      • BMO9 simulates a behavior (shared by 10 other holdouts[67]) similar to that of a stack,[68] whereas BMO10, a shift-overflow counter, also displays a behavior that resembles that of a one-dimensional cellular automaton.[69]
    • On August 3, prurq simulated three machines to halting.[70][71][72]
    • On August 4, q64 proved another machine halting.[73]
    • On August 15, Discord user jacobrschwartz proved another machine halting.[74]
    • On the same day, mxdys released a new holdouts list of 1,016 machines up to equivalence,[75] this is a 4.51% reduction compared to the previous holdouts lists.
    • On the same day, mxdys, using an LLM, produced proofs deciding 20 machines (the entirety of Counter Class 1, and 4 Dekaheptoids); 2 of these proofs (both belonging to Dekaheptoids) have been confirmed, whereas the other 18 proofs are under review.[76]
      • mxdys also says that the LLM has produced 32 other Rocq proofs, and has helped them find configurations in existing deciders that settle 10 other holdouts.[77]
    • On August 24, Discord user flintt_6 informally proved another machine halting.[78]
    • On the same day, mxdys released a list of machines that are easy to accelerate using the UBRRBA decider, consisting of 86 machines.[79]
    • On August 25, prurq simulated another machine to halting.[80]
    • On August 26, q64 proved another machine halting.[81]
    • On August 29, mxdys released another holdouts list of 1,003 machines up to equivalence.[82]
    • On August 31, mxdys released another holdouts list of 999 machines up to equivalence,[83] breaking the 1,000 holdout barrier.
  • BB(4,3)
    • Andrew Ducharme reduced the number of holdouts from 4,747,566 to 4,720,332 (a 0.57% reduction) via the mxdys FAR decider.[84]
  • 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.

Misc

  • On August 22, person89 began a discussion on what behaviors should be required to classify a machine, or another program, as a Cryptid.[85]
    • This may be relevant to the classification of BMO1, which is currently not considered a Cryptid,[86] despite it sharing many aspects with mathematically hard problems.
    • person89 has also suggested that machines or programs that would have relied on the validity of Fermat's Last Theorem (prior to Andrew Wiles's proof in 1994[87]) should be considered Cryptids in some sense, despite the problem being settled,[88] due to the length, complexity, and technicality of the proof.
    • Shawn Ligocki observed that the only programs that have been widely accepted as Cryptids are those that rely on Collatz-like randomness (e.g. Antihydra, Lucy's Moonlight).[89] They also brought up recent discussion about Diophantine Equations[90] about whether the smallest holdout equations can be considered Cryptids by virtue of being simple, open problems in number theory;[91] they believe otherwise, as not much effort has been put into deciding these holdouts.[92] (See also: discussion about whether x3+y3+z3=114 and other unsolved sums of three cubes problems are Cryptids). They also suggest that Cryptid-ness could instead considered as a spectrum, rather than a black-or-white category,[93] such as considering Fermat's Last Theorem as an easier type of Cryptid.[94]
    • It has been suggested that a program's Cryptid status should be determined by the complexity of its underlying behavior/problem, and whether it can be related to any mathematically hard problem.[95][96] Shawn Ligocki remarks that the main reason why other hard machines (e.g. Skelet 1, Skelet 17) should not be considered Cryptids is due to the lack of a simple restating of their halting requirements or a connection to a mathematically hard problem.
    • PAFs (Piecewise Affine Functions) and Collatz-like behaviors were noted to have a sharp increase in difficulty between currently solvable and unsolvable examples,[97] though star suggests that even 2-dimensional 2-regional PAFs have examples of medium-difficulty problems, and that they are likely Turing-complete.[98] She also notes how Collatz-like behavior seems to be common amongst the smallest holdouts across different BB-adjacent functions (e.g. Turing machines, Fractrans programs),[99] though there are likely some exceptions, such as the BBµ(16) chaotic function and the BB(3,3) holdout 1RB2LC1RC_2LC---2RB_2LA0LB0RA (bbch), as pointed out by Shawn Ligocki and LegionMammal978 respectively.[100][101]
  • On August 26, XnoobSpeakable began a discussion on archiving Discord messages which are used as sources on this wiki.[102]
    • Katelyn Doucette comments that not only does an archive provide the comfort of not losing valuable information should messages be taken down, it also provides an easier way to search through messages, contrasting Discord's suboptimal search feature.[103] She also provided a tool for mass archiving Discord messages without breaking its Terms of Service.[104]
    • cosmo notes that an indiscriminate public archive of the server had previously been suggested[105] and rejected[106] by members, as it would have changed how they would have interacted with others.[107] However, they do still keep a private, up-to-date archive of the whole server.[108]
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