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

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.

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(6): 15 solved machines. 1311 holdouts (3 informally solved this month, one informally solved earlier).
  • Progress has been made in reducing the list of machines not simulated up to 1e13, by Alistaire: see spreadsheet. Current count: 210. (Total reduction: 68 (+1 later). 38 (+1) machines^{[1][2][3][4][5][6][7][8][9][10][11][12][13][11 more][4 more][10 more][11 more][17 more]} simulated out, plus 2 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 a machine 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).
  • 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]