TMBR: March 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 Month in Beaver Research for March 2026. We celebrated bbchallenge's fourth birthday on 8 March. This month was quite a substantial month in Beaver research, as after more than 200 days, the last 3 informally solved BB(3,3) holdouts were formalised by mxdys in Rocq. Following the now-tradition, 3 new BB(2,5) machines were proven to not halt! This leaves us with 60 informal holdouts. There has been a 4.37% reduction in BB(6), and quite a lot of results in Fractran: With the help of AI Agent Claude Opus 4.6, it was proven tentatively that BBf(21) = 31,957,632 and that a Cryptid exists in BBf(22), which means that BBf(22) Is Hard.

Champions

• Discord user 50_ft_lock found a new BB(13) champion which surpasses Graham's number, reducing the upper bound of Graham-beating TMs to 13 states.

Misc

• 1RB0RB_1LC1RE_1LF0LD_1RA1LD_1RC1RB_---1LC (bbch) was shown to be a probviously non-halting Cryptid similar to Space Needle by DrDisentangle.
• mxdys made a new Turing Machine Visualizer using longitudinal acceleration (for shift-overflow mixed-digit non-unary counters)[1] which is available at https://ccz181078.github.io/TM/LongAcc/index.html

Meta

• Famous math Youtuber 2swap made a couple of videos about Turing Machines arranged into grids and colored based on their halting status for BB(2,2), BB(3,2), BB(2,3), BB(4,2) and BB(5,2) respectively, then made a two-hour long Youtube video about the same topic on their second channel.

BB Adjacent

• Fractran: In BBf(21), Claude Opus 4.6 gave a proof that all 140 holdouts do not halt. This tentatively proves that BBf(21) = 31,957,632.
• Fractran: A Cryptid was discovered in BBf(22) with the help of Claude Opus 4.6.
• Fractran: Katelyn Doucette started working on a program to visualize fractran spacetime diagrams just like for TMs:
  • Frankenstein's Monster
  • BBf(20) champion
  • Hydra
  • Space Needle

Holdouts

BB Holdout Reduction by Domain

Domain	Previous Holdout Count	New Holdout Count	Holdout Reduction	% Reduction
BB(6)	1214	1161	53	4.37%
BB(7)	18,195,192	18,036,852	158,340	0.87%
BB(2,5)	72	69	3	4.17%
BB(2,6)	548,993	545,005	3,988	0.73%

• BB(6): 1e14 machines: 170. 1e15 machines: 237. 53 solved machines.
  • Holdouts lists for machines not simulated to 1e14 and 1e15 steps were created. The holdouts list counts were 178 and 252 respectively. See Spreadsheet for BB6.
  • Later, prurq found 10 more machines in the holdouts list that had previously been simulated to 1e15: thus the new 1e15 holdout count was 242.
  • Andrew Ducharme solved two machines using FAR.
  • prurq found two machines [2][3] to be Translated Cyclers. Shawn Ligocki verified one of them and discovered its preperiod to be over $10^{12}$ steps. mxdys verified the other. Both were 1e14 and 1e15 holdouts, thus reducing those holdout counts by 2.
  • mxdys found four more [4][5] Translated Cyclers in the remaining holdouts and solved one more TM using FAR.
  • Andrew Ducharme solved a machine using FAR. This machine was a 1e14 and 1e15 holdout, thus reducing those holdout counts by 1.
  • prurq found 3 more machines previously simulated so far in the 1e14 list, and 2 more in the 1e15 list (both were also 1e14). This means 1e14 holdout count was reduced by 3, and 1e15 holdout count was reduced by 2.
  • Discord user mammillaria simulated a 1e14 holdout thus far, and then another 9 days later, therefore reducing that holdout count by 2.
  • prurq found a machine to be a Translated Cycler.
  • mxdys decided a machine using FAR.
  • mxdys released a new holdout list of 1161 machines. The new informal holdout count is 1159, and the Rocq-verified holdout count is 1187.
• BB(7):
  • Andrew Ducharme reduced the number of holdouts from 18,195,192 to 18,036,852 (a 0.87% reduction) via the mxdys FAR decider.
• BB(2,5):
  • Peacemaker II solved a machine using FAR and mxdys confirmed two[6][7] machines to be non-halting. Thus, the new holdout count is 69, or 60 considering informal proofs.
• BB(2,6):
  • A new filtering run by Andrew Ducharme has reduced to number of holdouts from 548,993 to 545,005.[8]
• BB(3,3):
  • mxdys formalised three remaining informal results (650, 412, 279) into Rocq.