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 next 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.
  • @mxdys shared a new way of grouping TMs by equivalence class, this reduced the number of holdout equivalence classes from 2467 to 1691 (31% reduction).
  • @mxdys published a list of 3171 holdouts. The holdouts are in 1691 equivalence classes by halting status.
• BB(7):
  • Andrew Ducharme has continued reducing the number of holdouts, from 59,727,905 to 28,189,617 (52.80% reduction).
  • Terry Ligocki ran an additional 41 filters/parameters. This reduced the number of holdouts, from 28,189,617 to 23,314,388 TMs (17.29% reduction)
  • The overall reduction this month was 72.9% (note: This point can be discarded after the table is made for overall reduction for all domains)
• BB(2,6):
  • @Peacemaker II noticed some TMs missing from Andrew Ducharme reductions. It was decided to start back at the original 22,302,296 holdout TMs. (TODO: Source)
  • Andrew Ducharme ran several deciders/filters and reduced the 22,302,296 holdout TMs to 20,358,011 (8.72% reduction). (TODO: Source)
  • Terry Ligocki ran 50 variations of deciders/parameters using @mxdys' C++ code, main.cpp, reducing the holdout count to 970,101 TMs (95.23% reduction)!
  • Andrew Ducharme ran some additional deciders reducing the holdout count to 873,469 TMs (9.96% reduction). (TODO: Link to wiki page where you can find the specific filters)
• BB(4,3):
  • Terry Ligocki ran a set of deciders/parameters from @mxdys' code to reduce the number of holdouts which were at 460,916,384 TMs. Six passes have reduced the holdout count to 97,701,052 TMs (78.80% reduction). (TODO: Source)
• BB(3,4):
  • TODO: XnoobSpeakable and Lúkos are working on the holdout list for BB(3,4), ~435M TMs, using @mxdys' deciders with various parameters.

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.
• 23 Sep 2025. Katelyn Doucette. Building the Busy Beaver Ladder.

Misc

• TODO: Affine maps are Turing complete and its relation to BMO1: https://discord.com/channels/960643023006490684/1239205785913790465/1420457986564030641 https://discord.com/channels/960643023006490684/1239205785913790465/1420491357969059910