TMBR: August 2026

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This Month in Beaver Research

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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

• 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]
    • 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]
• TT (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 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 9-pebble automaton running for 63,577 steps by modifying mathmasterzach's 323 step 9-pebble automaton. The first 200 steps of this champion are displayed here, with colors corresponding to the numbers of pebbles in a single cell. (Image source: 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 improved the peBBle(9) champion, creating a new automaton which runs for 63,577 steps.^[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; tiling rules are denoted by colors, similar to other depictions of Wang tiles. (Image source: 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--- (1LD1RE_0LA0RB_0RD0LD_0LE---&status=halt bbch), is shown here in mxdys's Turing Machine 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.