TMBR: December 2025

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This Month in Beaver Research

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This Month in Beaver Research for December 2025. In commemoration of the end of the year, we've summarized some of the major accomplishments into This Year in Beaver Research: TYBR: 2025. This month saw a lot of exploration into BB-adjacent models with the introduction of BB for General Recursive Functions and Register machines and new champions for many previously introduced variants.

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

• 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(91) champion, running for over $f_{{\epsilon }_0+1}\left(3\right)$ steps. f represents the Fast-Growing Hierarchy.
• On 11 Dec, a BB Fractran champion demonstrating $BBf(22)>10^{62}$ was found by Shawn Ligocki^[1] and verified by Jason Yuen.^[2] This was the first Fractran champion which required accelerated simulation. On 12 Dec, Shawn modified that program to create Frankenstein's Monster, a BBf(23) Cryptid.^[3]
• 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.
• Azerty discovered a series of Blanking Busy Beaver 3x3 champions, culminating in 1RB2LC2LA_1LC---2RA_2RC2LB0LC (bbch) which blanks the tape after 329 steps on 31 Dec 2025.
• 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.
• 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.
• Azerty shared a website with champions for many BB-Adjacent functions and original BB.

BB Adjacent

• Many new deciders were introduced for Fractran. BBf(20) = 746 was solved. BBf(21) holdouts were reduced from 9427 to 345. BBf(22) was enumerated and reduced to 5682 holdouts. A Cryptid was constructed in BBf(23).
• Busy Beaver for General Recursive Functions (BBµ) was introduced by Shawn Ligocki, computed up to BBµ(3) = 1 and general lower bound $BB\mu (6k+19)\ge 2{\uparrow }^{k}5$ was established.
• Busy Beaver for Register machines (MBB(n)) was introduced and computed up to n = 5 and a lower bound for n = 6 was discovered.
• Busy Beaver for lambda calculus using De Bruijn indexes was introduced with lower bounds having been calculated up to n = 26.

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. (That is, not verified in Rocq.)
  • 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.

Misc

• kalikimaka sonified Antihydra.^[4]