TMBR: December 2025

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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).

This is the last edition of TMBR this year. 2025 was a very productive year for BBChallenge: about 60% of the next domain, BB(6), was solved. Furthermore, new champions were discovered for BB(6), BB(7) and BB(4,3). Many models of computation other than Turing Machines were also explored - most notably Fractran and Instruction-Limited Busy Beaver. Some new methods were developed, such as mxdys's new version of FAR.

This year, Themed Months were introduced - first, for BB(3,3), then for BB(2,5) - and the result is the clarification and verification of some of the results and techniques on the Discord and wiki. See TMBR: November 2025#Themed Months for more information.

An annotated spreadsheet of BB(6) holdouts was also shared by Robin Rovenszky, which includes links to Discord discussions, classification of machines and is almost always up-to-date. See Google Sheets

This Year in Beaver Research _{(TYBR - "Thank You Beaver Researchers!")}

Holdouts Reductions.

• BB(6) - Reduced from 3571 to 1326 holdouts. Hence, 2245 machines were solved this year. This is a 63% reduction.
• BB(2,5) - Reduced from 217 to 75, a 65.43% reduction. (The number of informal holdouts is 64).
• BB(7) - Enumeration was completed, the number of holdouts was reduced from an initial 85,853,789 to 20,387,509 machines, a 76.25% reduction.
• BB(4,3) - Reduced from 460,916,384 to 9,401,447 holdouts, a 97.96% reduction.
• BB(3,4) - Reduced from 434,787,751 to 12,435,284 holdouts, a 97.14% reduction.
• BB(2,6) - Enumeration was completed, the number of holdouts was reduced from an inital 2,278,655,696 to 870,085 machines, a near 100% reduction.
• BB(2,7) - Enumeration started, 100K of the 1M subtasks have been enumerated (10%).

Champions.

• BB(6) - On 16 June 2025, mxdys discovered 1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB (bbch), running for 10 ↑↑ 11010000 steps. This was surpassed on 25 June when mxdys discovered 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch), a TM which runs for $10\uparrow \uparrow 10\uparrow \uparrow 10\uparrow \uparrow 8.10237$ steps.
• BB(2,5) - The champion, initially discovered by Daniel Yuan on 24 Jun 2024 was verified by mxdys on 4 Jun 2025.
• BB(7) - Within three days of the start of the enumeration of BB(7), three champions were discovered. The first two were discovered by Shawn Ligocki: 1RB0RF_1LC0RE_1RD1LB_1LA1LD_0RA0LE_1RG0LB_1RZ1RB (bbch) with a sigma score of about 10 ↑↑ 22 and 1RB1RA_1RC0LC_0LD1LG_1LF0LE_1RZ1LF_0LA1LD_1RA1LC (bbch) with a sigma score of about 10 ↑↑ 35. This was followed by the discovery of 1RB0LG_1RC0RF_1LD1RZ_1LF0LE_1RA1LD_1LG1RE_0LB0LB (bbch), achieving a sigma score of about 10 ↑↑ 46, by Terry Ligocki. On 10 May 2025, Pavel Kropitz discovered 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch), a TM which runs for over $2{\uparrow }^{11}2{\uparrow }^{11}3$ steps.
• BB(4,3) - In Feb 2025, Racheline identified 0RB1RZ0RB_1RC1LB2LB_1LB2RD1LC_1RA2RC0LD (bbch) as the new BB(4,3) champion with a score of over $2\uparrow \uparrow \uparrow 2^{2^{32}}$. In Oct 2025 Polygon identified a new BB(4,3) and num(4,3) champion with a score of over $10{\uparrow }^{4}4$ (1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)). These TMs were first proven to halt by Pavel Kropitz in May 2024, but their runtimes were not known at the time.
• BBB(3,3) - In March 2025 Nick Drozd discovered 1RB0LB2LA_1LA0RC0LB_2RC2RB0LC (bbch), which quasihalts after running for more than 10 ↑↑ 6 steps.
• Busy Beaver for lambda calculus - The values of BBλ(31), BBλ(32), BBλ(33), BBλ(34), BBλ(35), BBλ(36) and BBλ(37) were solved, while new champions for BBλ(40), BBλ(41), BBλ(42), BBλ(43), BBλ(44), BBλ(45), BBλ(46), BBλ(48), BBλ(63) and BBλ(91) were discovered.
• BBS(4,3) - A new champion (1RB1RD1LC_2LB1RB1LC_1LB1LA1LD_0RB2RA2RD (bbch)) was discovered by changing the C0 transition of the BB(4,3) champion 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch) from C0 --> 1RZ to C0 --> 1LB.
• BLB(3,3) - On 27 Dec 2025, Azerty discovered 1RB1LA---_2RC2LB1RB_2LC2LA0RC (bbch) which blanks the tape after 225 steps. A day later he found 1RB1RC---_1LB1RA2RB_0RB2LC0RC (bbch), which blanks the tape after 308 steps. That record was surpassed again on 31 Dec 2025 by the discovery of 1RB2LC2LA_1LC---2RA_2RC2LB0LC (bbch) which blanks the tape after 329 steps.
• BBP(3,3) - On 25 Dec 2025, Azerty discovered 1RB2RC1LC_0RC0RB1LA_2LA2RC1LB (bbch) which is a Translated cycler with a new record period length of 1195 steps.

New Methods.

• New FAR using DFA generator by mxdys.^[1][2]
• @Bricks shared a method to estimate susceptibility to Block Analysis and a spreadsheet of BB(6), BB(3,3) and BB(2,5) holdouts quantified by it.^[3][4]

TODO: Before July

Misc.

• A fast algorithm for Consistent Collatz simulation was re-discovered and popularized. Using it,
  • apgoucher simulated Antihydra to $2^{38}$ iterations. This is actually a result from one year ago, but was rediscovered and added to the wiki. Source
  • Shawn Ligocki simulated 1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC (bbch) out to one additional Collatz reset, demonstrating that (if they halt, which they probviously should) they will have sigma scores $>10^{10^{10^7}}$.
  • This algorithm has near linear runtime (in the number of iterations simulated), but also linear memory growth since the parameters grow exponentially. This memory limit seems to be the main bottleneck to simulating Antihydra and other Consistent Collatz iterations further. There has been some discussion on more efficient memory usage or a distributed algorithm to support further scaling, but no results are available yet.
• Andrew Wade claims to have proven that BB(432) is independent of ZF. Source
• Piecewise Affine Functions (PAF) were explored as a generalization of the BMO1 rules:
  • @Bard proved that 3 dimension PAF are Turing complete.^[1]
  • @star proved that 2 dimension PAF are Turing complete.^[2][3]
  • Shawn Ligocki wrote up a proof sketch that 2-region PAF are Turing complete.^[4]
  • It was discovered that Amir Ben-Amram had already proven both of these results in 2015 (both the 2-dim and the 2-region results).
  • BMO1 is a 2-dim, 2-region PAF so this provides some sense for the difficulty of the problem.
  • This introduces a new type of Cryptids separate from previous Collatz-like ones.
• @coda shared a mechanical implementation of Antihydra^[5] and @zts439 3d-printed a prototype.^[6]
• @vonhust created a fast TM simulator that averages 2 billion steps / s. It uses fixed-block Macro Machines with each block bit-packed into integers. It is about 10x faster than direct simulators across most TMs.^[7]

TODO: Before July

BB Adjacent.

• Instruction-Limited Busy Beaver was introduced and calculated up to BBi(7).
• Reversible Turing Machine Busy Beaver values were calculated up to BB_rev(5).
• Terminating Turmites (Relative Movement Turing Machines) were introduced and lower bounds for some small domains have been computed.
• John Tromp introduced the $BB{\lambda }_1(n)$ function for Busy Beaver for lambda calculus with an oracle and computed it up to $BB{\lambda }_1(22)$.
• Instruction-Limited Greedy Busy Beaver gBBi(n) and an Instruction-Limited variant of the Blanking Busy Beaver (BLBi(n)) were introduced. gBBi(n) was computed up to n = 13 and BLBi(n) was computed up to n = 7.
• @savask shared the Bug Game (and fast-growing $Bug(H,W)$ function).
• Busy Beaver for Fractan (BBf) was introduced on 1 Nov by Jason Yuen.^[8] Exact values have been proven up to BBf(20) = 746^[9][10] and exhaustive enumeration has been run up to size 22 (with BBf(21) > 31,957,632 and 397^[11] holdouts, and $BBf(22)>1.146\times 10^{62}$ and 7807^[12] holdouts).
• Cyclic Tree Busy Beaver (CTBB) was introduced by @Jack on 14 Nov.^[13] The exact value is known for CTBB(2) = 5 and lower bounds have been found up to size 7 with CTBB(7) > 4↑↑↑↑(4↑↑↑3).
• Busy Beaver for lambda calculus using De Bruijn indexes was introduced with lower bounds having been calculated up to n = 34.

TODO: Before July and Semi-infinite tape Busy Beaver https://discord.com/channels/960643023006490684/1450894930422530161/1450894930422530161

In the News.

• 6 January 2025. It Boltwise. Durchbruch im Busy Beaver Problem: Eine neue Ära der Mathematik (German) (English: Breakthrough in the Busy Beaver problem: A new era of mathematics).
• 24 March 2025. Nick Drozd. BBB(3, 3) > 10 ↑↑ 6.
• 21 Apr 2025. Shawn Ligocki. Lucy's Moonlight: The 5% Champion.
• 9-13 June 2025. Terence Tao mentioned bbchallenge in their talk "The Equational Theories Project: advancing collaborative mathematical research at scale" (video / slides) at the 2025 Big Proof workshop. The talk is about the Equational Theories Project, a large-scale mathematical collaboration that crowd-sourced a proof in Lean. Tao mentions bbchallenge as the only other example of a large-scale mathematical collaboration to prove a single result that he knows of.
• 28 June 2025. Scott Aaronson. BusyBeaver(6) is really quite large.
• 1 July 2025. The Quanta Podcast. How Amateurs Solved a Major Computer Science Puzzle.
• 2 July 2025. Manon Bischoff. Spektrum. Wie der sechste Fleißige Biber die Mathematik an ihre Grenzen bringt.
• 3 July 2025. Nick Drozd. Busy Beaver Backwards.
• 7 July 2025. Karmela Padavic-Callaghan. New Scientist. Mathematicians are chasing a number that may reveal the edge of maths. (Paywalled)
• 9 July 2025. David Roberts. BB(5)=47,176,870: BB(6) is … astronomically larger.
• 11 July 2025. New Scientist podcast episode 311. Discusses mxdys's BB(6) pentation result "We’re brushing up against the edge of mathematics".
• 11 July 2025. Darren Orf. Popular Mechanics. Mathematicians Say There’s a Number So Big, It’s Literally the Edge of Human Knowledge.
• 14 July 2025. Joe Brennan. Dario AS. Meet the Busy Beaver number, a number so huge that mathematicians call it the frontier of mathematical knowledge
• 15 July 2025. Nick Drozd. Performance Hacks for Brady's Algorithm.
• 18 July 2025 https://francis.naukas.com/2025/07/18/espeluznante-nueva-cota-inferior-para-la-funcion-castor-afanoso-bb6/
• 22 Aug 2025. Ben Brubaker. Quanta Magazine. Busy Beaver Hunters Reach Numbers That Overwhelm Ordinary Math.
• 25-29 Aug 2025. Tristan Stérin presented a poster at DNA 31.
• 1 Sep 2025. Katelyn Doucette. All About Space Needle.
• 12 Sep 2025. Katelyn Doucette. Bugs, Mazes, and the Unreasonably Effective Brady's Algorithm.
• 14 Sep 2025. Ben Brubaker. Wired. The Quest to Find the Longest-Running Simple Computer Program. (Reprint of Quanta article from last month).
• 17 Sep 2025. Hacker News. Determination of the fifth Busy Beaver value.
• 18 Sep 2025. Tuomas Kangasniemi. Tekniikkatalous. Iso matematiikan ongelma ratkesi 63 v jälkeen (Finnish) (English: A big math problem solved after 63 years).
• 23 Sep 2025. Katelyn Doucette. Building the Busy Beaver Ladder.
• 30 Sep 2025. Nick Drozd. The Shape of a Turing Machine.
• 22 Oct 2025. Ben Brubaker. Why Busy Beaver Hunters Fear the Antihydra. (Hacker News thread)
• 27 Oct 2025. Tristan Stérin gave a talk about bbchallenge and the BB(5) proof at Collège de France: Le cinquième nombre Busy Beaver (in French).^[1]
• 7-9 Nov 2025. Carl Kadie gave a talk on BB during the PyData Seattle 2025 conference: How to make Python programs run very slow (and new Turing Machine results).^[2]
• 26 Dec 2025. New Scientist. Mathematicians spent 2025 exploring the edge of mathematics. (Paywalled)
• 31 Dec 2025. Nick Drozd. Running out of places to move the goalposts to.

TODO: Before July

Misc

• Azerty shared a website with champions for many BB-Adjacent functions and original BB.

In the News

• 26 Dec 2025. New Scientist. Mathematicians spent 2025 exploring the edge of mathematics. (Paywalled)

Blog Posts

• 31 Dec 2025. Nick Drozd. Running out of places to move the goalposts to.

Champions

• A new BBS(4,3) champion (1RB1RD1LC_2LB1RB1LC_1LB1LA1LD_0RB2RA2RD (bbch)) was discovered by changing the C0 transition of the BB(4,3) champion 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch) from C0 --> 1RZ to C0 --> 1LB.
• On 27 Dec 2025, Azerty discovered 1RB1LA---_2RC2LB1RB_2LC2LA0RC (bbch) which blanks the tape after 225 steps. A day later he found 1RB1RC---_1LB1RA2RB_0RB2LC0RC (bbch), which blanks the tape after 308 steps. That record was surpassed again on 31 Dec by the discovery of 1RB2LC2LA_1LC---2RA_2RC2LB0LC (bbch) which blanks the tape after 329 steps.
• On 25 Dec 2025, Azerty discovered 1RB2RC1LC_0RC0RB1LA_2LA2RC1LB (bbch) which is a Translated cycler and new BBP(3,3) champion with a new record period length of 1195 steps.
• Patcail and Gustavo Melo discovered a new BBλ(63) champion, running for more than $f_{{\omega }^3}\left(2\right)$ steps and 50_ft_lock discovered a new BBλ(92) champion, running for over $f_{{\epsilon }_0+1}\left(3\right)$ steps. f represents the Fast-Growing Hierarchy.
• On 18 Dec Azerty discovered the new TT(2) champion 1TB---_1PA0PB with a runtime of 13 steps. This was followed by new champions a day later: 1PB0PA_1TA0PC_1PA--- for TT(3) with a runtime of 82 steps, 1TB---_0PD1PB_1PA1TA_0PC0PD for TT(4) with a runtime of 758 steps, 1TB0PA2PA_2PA---1PA for TT(2,3) with a runtime of 223 steps and 1TB3TB2PB---_2TB1PA0PA2TB for TT(2,4) with a runtime of 1,068 steps. On 21 Dec Azerty also discovered the new TT(3,3) champion with a runtime of 427 steps 1TB2TA2PA_1TA0TC1PC_---2PA0PA, this champion was surpassed on 22 Dec by 1TB2PB1PB_2TA0TA2PC_2TA---2PA with a runtime of 1,072 steps, on 23 Dec by 1PB2PC1PB_2TC0TA---_1PA1PC0PC with a runtime of 3,786 steps and again on 28 Dec by 1PB1PA1TA_2TB2PB2PC_---2PA1TC with a runtime of 45,153 steps.
• num TODO

BB Adjacent

• Busy Beaver for lambda calculus using De Bruijn indexes was introduced with lower bounds having been calculated up to n = 34.

TODO. Register machines, General Recursive Functions, Fractran progress, Semi-infinite tape Busy Beaver

Holdouts

BB Holdout Reduction by Domain

Domain	New Holdout Count	Previous Holdout Count	Holdout Reduction	% Reduction
BB(6)	1326	1416	90	6.36%
BB(7)	20,387,509	20,405,295	17,786	0.09%
BB(3,4)	12,435,284	15,136,283	2,700,999	17.84%

• BB(6): 90 machines solved, a 6.36% reduction.
  • There are 14 holdouts left to simulate up to 1e12 steps, and 288 to simulate up to 1e13 steps^[1]. Later, at the end of the year, all 14 machines were simulated up to 1e12 machines, which means that now, all holdouts TMs must have a score >1e12. 278 machines are left to simulate up to 1e13. A dynamic list can be found here. (See results on Discord, plus [2], [3] and [4])
  • The possibility of simulating computationally tractable machines which nonetheless has large time and memory requirements was discussed. List
  • mxdys shared a new holdouts list, consisting of 1343 machines, which means 73 solved TMs. This is a 5.4% reduction. There is one extra machine that is solved formally, but unverified.
  • The old spreadsheet was replaced with a newer one, see Google Sheets
  • At the end of the year, mxdys shared the latest holdouts list, which consisted of 1326 machines, that is, 17 TMs solved and a 1.27% reduction.
• BB(7):
  • Further enumeration by Andrew Ducharme has reduced the number of holdouts from 20,405,295 to 20,387,509, a 0.09% reduction.
• BB(3,4):
  • XnoobSpeakable and Lúkos ran stages 8, 9, 10A, 10B & 10C of Phase 2, reducing the number of holdouts from 15,136,283 TMs to 12,435,284 holdouts. This is a 17.84% reduction.
• BB(2,7):
  • Terry Ligocki enumerated 60K more subtasks, increasing the number of holdouts to 309,868,865. A total of 100K subtasks out of the 1 million subtasks (or 10%) have been enumerated.