TMBR: September 2025

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This Month in Beaver Research

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This Month in Beaver Research for September 2025.

TODO: BB(5) arXiv released https://arxiv.org/abs/2509.12337

TODO: BB(3,3) month.

In the News

• 14 Sep 2025. Ben Brubaker. Wired. The Quest to Find the Longest-Running Simple Computer Program.
• 17 Sep 2025. Hacker News. Determination of the fifth Busy Beaver value.
• 18 Sep 2025. Tuomas Kangasniemi. Tekniikkatalous. Iso matematiikan ongelma ratkesi 63 v jälkeen (Finnish) (English: A big math problem solved after 63 years).

Holdouts

• BB(6):
  • Andrew Ducharme and @mxdys both found a family of 10 halting TMs independently, all halting in around $10^{69}$ steps.
  • @mxdys decided two machines’ fates from the 50 Random Holdouts released in August, making 8/50 machined solved as of this month.
  • Andrew Ducharme found a non-halting TM, to which Peacemaker II replied with another machine that turned out to be very similar, and non-halting as well. Later in his search Andrew found another non-halting TM.
  • TODO: mxdys decided 776 holdouts using a new equivalence
• BB(7):
  • Andrew Ducharme has continued reducing the number of holdouts, from 59,727,905 to 28,189,617 (52.80% reduction)
  • TODO: Terry Ligocki is running an additional 41 filters/parameters. 15 filters have been completed which reduced the holdouts from 28.2M to 24.9M (11.64% reduction).
• BB(2,6):
  • Andrew Ducharme has continued reducing the number of holdouts from 18,054,938 to 17,629,828 (2.35% reduction).
  • Terry Ligocki shared that he ran 14 of @mxdys deciders on the holdouts list, reducing the holdout count to a flattering 1,300,334 TMs (92.63% reduction).
  • @Peacemaker II discovered there were TMs missing from Andrew Ducharme's initial reductions. Andrew decided to scrap those reductions and run a single filter and reduced the holdout TMs from 22.3M to 20.4M (8.52% reduction).
  • Terry Ligocki then reran the 14 filters on the corrected holdouts and reduced the holdouts from 20.4M to 1.43M (92.9% reduction). This was followed by running 36 filters found examining a subset of the holdouts which reduced the holdouts from 1.43M to 970K (32.2% reduction). Overall the 50 filters reduced the holdouts by 95.23%.
  • TODO: Andrew Ducharme is continuing to run filters and has gotten some more appreciable reductions.

BB Adjacent

• John Tromp announced on Discord that a 350-bit $BB\lambda (n)$ function now reaches the limit of BMS, an improvement from the previous 404 bits.

TODO: phrasing. Discord source: https://discord.com/channels/960643023006490684/1355653587824283678/1413637783045542038 and https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/bms.lam

• TODO: savask shared "Busy Bug Game": https://discord.com/channels/960643023006490684/1362008236118511758/1415723582989930679

Blog Posts

• 12 Sep 2025. Katelyn Doucette. Bugs, Mazes, and the Unreasonably Effective Brady's Algorithm.
• 15 Sep 2025. Katelyn Doucette. Building the Busy Beaver Ladder.