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.
  • At the start of the month, 14 holdouts had not been simulated to 1e12 steps and 288 holdouts had not been simulated to 1e13 steps. This month, all 14 of the former holdouts and 10 of the latter holdouts were simulated past 1e12 steps and 1e13 steps respectively. This means all holdout TMs must continue for at least 1e12 steps. 278 machines are left to simulate up to 1e13 steps. A dynamic list can be found here. (See results on Discord, plus [1], [2], [3] and [4])
  • The possibility of simulating computationally tractable machines which nonetheless has large time and memory requirements was discussed. See a list of such TMs here.
  • 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, meaning the solution of 17 TMs 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]