TMBR: January 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).

This is the first edition of TMBR in 2026.

TODO: mxdys released C++ implementation of FAR decider on 1/18, also add Code repositories page (is the FAR decider present there?)

Champions

• nickdrozd discovered a new BLB(3,3) champion (1RB2RB1LA_2LC0LB2LB_2RC2RA0LC (bbch)) which blanks the tape after running for more than 10^42,745 steps.
• creeperman7002 discovered 1TB1PA_1PC0PA_1TA0PD_---1TA, a TT(4,2) TM which runs for 48,186 steps and 1TA2PB3TB---_3TA1PB1TA1PA, a TT(2,4) TM which runs for more than 3.467*10¹⁵ steps.
• A new lower bound of $f_{{\epsilon }_0{\omega }^{{\omega }^3}}\left(4\right)$ was computed for the BBλ(91) champion.
• New champions were discovered for BBλ(61), BBλ(86), BBλ(90), BBλ(94), BBλ(96) and BBλ(100).

Blog Posts

• 16 Jan 2026. Nick Drozd. AI Reproduction of Lin's Busy Beaver Proof.
• 28 Jan 2026. John Tromp. The largest number representable in 64 bits - Revised.

BB Adjacent

• Uniform-Action Busy Beavers were introduced, and lower bounds have been given up to BBu(6) (with BBu(2)=6, BBu(3)=17 and BBu(4)=29)

Theory

@ConePine shared an idea on how the value of BB(5) can be proved without enumerating Turing machines.

Holdouts

BB Holdout Reduction by Domain

Domain	New Holdout Count	Previous Holdout Count	Holdout Reduction	% Reduction
BB(2,5)	74	75	1	1.3%
BB(6)	1314	1326	12	0.9%
BB(7)	19,303,801	20,387,509	1,083,708	5.32%
BB(2,6)	558,039	870,085	312,046	35.86%

• BB(6): 15 solved machines. 1311 holdouts. 2 informally solved this month.
  • Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire and aparker: see spreadsheet. Current count: 205. (Total reduction: 73 (+1 later). 39 machines^{[1][2][3][4][5][6][7][8][9][10][11][12][13][11 more][4 more][10 more][11 more][17 more][aparker's machine]} simulated out, plus 6 solved machines).
  • Alistaire found a halting machine in the list mentioned above, see Discord. Approximate score: 4e12. Later, he found another halting machine in the same list, see Discord - approximate score: 1.5e18.
  • @mxdys shared a list of machines that seem to be provable. See Discord.
  • Dyuan shared an informal proof for two machines nonhalting, using Longitudinal Analysis.
  • @Peacemaker II found two machines to halt with a bespoke accelerated simulator.
  • @mxdys released a holdouts list of 1314 machines up to equivalence, which is a 1% reduction (12 machines) from last month. 4 of these were not previously simulated up to 1e13 steps, thus that count has been lowered to 206.
  • @mxdys announced the start of translating three proofs for halting machines into Rocq (two machines found by @Peacemaker II this month, one proof for a machine found to halt by Racheline in July 2024, which has previously been missed when accounting for the informal holdout count, therefore the informal holdout count is now one less).
  • All machines of the "Unknown" class have been simulated up to 1e13 steps, with aparker simulating the last machine to 1.15e13: Discord.
• BB(7):
  • Andrew Ducharme further continued reducing the number of holdouts, from 20,387,509 to 20,197,978 TMs, a 0.93% reduction. This puts the total compute time on the current BB(7) pipeline just over 50,000 hours.^[1]
  • The holdout count was further reduced to 19,879,953 (a 1.57% reduction), which breaks the 20 million barrier.^[2]
  • Another reduction brought the holdout count down to 19,781,295, a 0.5% reduction, then 19,303,801, a 2.41% reduction.
• BB(2,5):
  • Andrew Ducharme found a machine to be nonhalting via @mxdys's FAR decider, which was confirmed in Rocq.
• BB(2,6):
  • Andrew Ducharme reduced the number of holdouts from 870,085 to 867,008, a 0.35% reduction, with more application of Enumerate.py.
  • Using the newly released mxdys FAR decider, the holdout count was brought down to 558,039, a 35.6% reduction.[3]