TMBR: December 2025

Prev: November 2025

This Month in Beaver Research

Next: January 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).

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.

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

Holdouts Reductions.

• BB(6) - Reduced from 3571 to 1416 holdouts. Hence, 2155 machines were solved this year. This is a 60% reduction.
• BB(2,5) - Reduced from 217 to 75, a 65.43% reduction.
• BB(7) - Enumeration was completed, the number of holdouts was reduced from an initial 85,853,789 to 20,405,295 machines, a 76.23% 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 14,518,243 holdouts, a 96.66% reduction.
• BB(2,7) - Enumeration started, 50K of the 1M subtasks have been enumerated (5%).

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) - Polygon identified a new BB(4,3) champion with a score of over $10{\uparrow }^{4}4$ (1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)). This TM was first proven to halt by Pavel Kropitz in May 2024, but its runtime was not known at the time.

New Methods.

• New FAR using DFA generator by mxdys.^[1][2]

TODO: Before July and this list:

1. TMBR: October 2025#Misc (Method to measure susceptibility to block-analysis)

Misc.

TODO: Before July and this list:

1. TMBR: August 2025#Cryptids (Fast algo for Consistent Collatz, BB(432) independent of ZF)
2. TMBR: October 2025#Theory (Piecewise Affine Functions)
3. TMBR: October 2025#Misc (Mechanical implementation of Antihydra)
4. TMBR: November 2025#Optimization (Vonhust's 2B steps/s simulator)

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.
• 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.^[1] Exact values have been proven up to BBf(19) = 370 and exhaustive enumeration has been run up to size 21 (with BBf(21) ≥ 31,957,632 and 587 holdouts).
• Cyclic Tree Busy Beaver (CTBB) was introduced by @Jack on 14 Nov.^[2] 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).

TODO: Before July

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).
• 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.
• 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).
• 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)

TODO: Before July + Talks(?):

1. DNA 31 - TMBR: August 2025
2. Three more in TMBR: November 2025#Talks

BB Adjacent

TODO. Register machines, General Recursive Functions, Fractran progress.

Holdouts

• BB(6):
  • There are 14 holdouts left to simulate up to 1e12 steps, and 312 to simulate up to 1e13 steps^[1]. The two lists can be found here.
• BB(3,4):
  • XnoobSpeakable continued reducing the number of holdouts with Stage 8 of Phase 2, by reducing it from 15,136,283 to 14,518,243 TMs. This is a 4.08% reduction.
• BB(2,7):
  • Terry Ligocki enumerated 10K more subtasks, increasing the number of holdouts to 150,662,006 and making 50K of the 1 million subtasks enumerated or 5%.