TMBR: August 2026

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This Month in Beaver Research

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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 × 10^9,595. 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.

General

• On August 2, @carrino, using Claude AI solved the last 290 holdouts, meaning BBB(4) is solved! Specifically, @carrino proved that BBB(4) = 32,779,478, the champion being 1RB1LC_1RD1RB_0RD0RC_1LD1LA (bbch), discovered by Nick Drozd in July 2021,^[1] 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 another unofficial holdout list.^[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, possibly meaning it has no stable orbits that would 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; another 4 holdouts have been removed due to equivalence, and another 3 holdouts have been removed due to implications. 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) being previously described as an ugly version of BMO1.^[8]
  • On August 22, @-d enumerated 20% of BBf(24), discovering a new champion, which runs for over 4.804 × 10^1,184 steps.^[9]
  • On the same day, Shawn Ligocki discovered a new BBf(24) champion which runs for over 9.263 × 10^9,595 steps.^[10]
  • On August 26, Shawn Ligocki discovered two new size 24 halters (non-champions) which have behaviors similar to Collatz-like and BMO1.^[11]
• 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.^[12]
  • 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.^[13]
  • On August 11, Discord user creeperman7002 released a TT(3,3) holdouts list consisting of 19,368 holdouts.^[14]
• SKI Calculus
  • On August 6, Δ⁵ discovered a new champion for Ξ₀_SK(16).^[15]
  • On the same day, Δ⁵ discovered a new champion for Ξ₀_BCKW(6).^[16]
  • On August 14, Δ⁵ discovered another new champion for Ξ₀_SK(16).^[17]
  • On August 20, Δ⁵ discovered new hexational champions for Ξ₀(17), Ξ₀(18), and Ξ₀(19) in the SKI calculus, and Ξ₀_SK(23), Ξ₀_SK(24), and Ξ₀_SK(25) in the SK calculus.^[18]
  • On the same day, Δ⁵ discovered another new champion for Ξ₀_SK(25).^[19]
• Lambda Calculus
  • On August 7, BBλ(41) was solved!^[20] We now know that BBλ(41) = 5 × 3³³ + 6 after solving 237 holdouts, thanks to the help of 2014MELO03, cppdecksucker, mxdys, and others.^[21]
    • The bet between Sam and cppdecksucker (which began in April^[22]) 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.^[23]

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.^[24]
  • On the same day, sheep discovered an infinite family of automata, including the champion for peBBle(4), showing peBBle(n) ≥ $\frac{n(n-1)}{2}$. ^[25]
  • On August 9, AlephSquirrel discovered a new champion for peBBle(5).^[26]
  • On August 10, Deccy discovered a new champion for peBBle(6)^[27]
  • On August 11, Deccy discovered a new champion for peBBle(7).^[28]
  • On August 13, mathmasterzach discovered 3 new peBBle(8) champions in quick succession,^[29][30][31] and the previous peBBle(9) champion, which runs for 323 steps.^[32]
  • On August 16, AlephSquirrel proved the peBBle(4) champion to be optimal.^[33]
  • 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).^[34]
• Diophantine Equation
  • On August 12, Shawn Ligocki extracted new bounds for BBdio(n) for heights 40, 48, 57, and 66, based on work done in the sums of three cubes problem.^[35][36]
  • On the same day, Shawn Ligocki suggested calling the equation $x^3+y^3+z^3=114$, and similar equations from the sums of three cubes problem, Cryptids,^[37][38] though their classification was later changed to that of energy vampire,^[39] following an objection by sheep;^[40][41] she suggested calling them "energy vampired [sic]",^[42] as these equations are expected to be solvable in practice, though they may be computationally hard to find solutions for.^[43]
  • On August 14, Shawn Ligocki rediscovered papers (and a post on StackExchange) from Bogdan Grechuk that imply BBdio(29) > 10⁶,^[44] and that BBdio(31) is decidable.^[45]
  • On August 23, star discovered the height 78 equation $a^4+b^4-c^4+30=0$, which is expected to not have any solutions, but may be very difficult to prove.^[46]. She also discovered the height 135 equation $w^5+x^5+y^5+z^5-7=0$,^[47] which is also expected to not have any solutions, but may be difficult to prove, 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.^[48]
  • On August 15, sheep made a size 35 program that tests the primality of a number.^[49]
  • On August 24, sheep discovered a set of Collatz-like programs,^[50] some of which simulate random unbiased walks.^[51][52][53] 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.^[54]

A new champion for BB_WT(9) can be obtained from a generalization of sheep's BB_WT(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 BB_WT(7) champion.^[55]
  • On the same day, Azerty implied the lower bound BB_WT(n) ≥ 3n - 10.^[56]

• Cyclic Tag
  • On August 26, Azerty introduced the Busy Beaver function for Cyclic Tag Systems, BBCT(n),^[57] proving the values of BBCT(n) for sizes 1 to 4, and providing lower bounds for BBCT(n) for sizes 1 to 12.^[58]
  • On the same day, sheep rediscovered literature that provided lower bounds for BBCT(n) for sizes 1 to 9.^[59]
  • On August 30, sheep discovered a size 36 Cryptid, showing BBCT(36) is hard.^[60]
• General Recursive Function
  • On August 27, Shawn Ligocki renewed interest in a size 16 chaotic (possibly Cryptid) GRF previously discovered in May,^[61] during a discussion on what behaviors should be required to classify a Cryptid.^[62]

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 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 BMO 9 and BMO 10 to the Beaver Math Olympiad.^[63][64]
  • On August 3, prurq simulated three machines to halting.^[65][66][67]
  • On August 4, q64 proved another machine halting.^[68]
  • On August 15, Discord user jacobrschwartz proved another machine halting.^[69]
  • On the same day, mxdys released a new holdouts list of 1,016 machines up to equivalence,^[70] 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, with 18 proofs under review at the time.^[71]
  • On August 24, Discord user flintt_6 informally proved another machine halting.^[72]
  • On the same day, mxdys released a list of machines that are easy to accelerate using the UBRRBA decider, consisting of 86 machines.^[73]
  • On August 25, prurq simulated another machine to halting.^[74]
  • On August 26, q64 proved another machine halting.^[75]
  • On August 29, mxdys released another holdouts list of 1,003 machines up to equivalence.^[76]
  • On August 31, mxdys released another holdouts list of 999 machines up to equivalence,^[77] 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.^[78]
• 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.