TMBR: October 2025

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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).

A brave busy beaver confronts the dreaded Antihydra. Copyright Nico Roper.

Misc

TODO: Add the mechanical Turing Machine. discord source: https://discord.com/channels/960643023006490684/1362008236118511758/1425889021653160026 and onwards

TODO: Add the method used by @Bricks to measure susceptibility to block-analysis. discord source: https://discord.com/channels/960643023006490684/1239205785913790465/1430227817957953638 also results: https://docs.google.com/spreadsheets/d/1j00LBxxp9W7uz1wZdMIvDCZ56eReuH0IGO9Z8-yybcQ/edit?usp=sharing

Wily Coyote, a BB(3,3) holdout

Blog Posts

• 22 Oct 2025. Ben Brubaker. Why Busy Beaver Hunters Fear the Antihydra.
• ? Oct 2025. Katelyn Doucette. Building the Busy Beaver Ladder.

Champions

• BB(4,3):
  • Polygon analysed the remaining "potential champions" discovered by Pavel Kropitz in May 2024, discovering that 1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch) is the new BB(4,3) champion with a score of over $10{\uparrow }^{4}4$.

Holdouts

• BB(6):
  • @mxdys shared a new holdouts list on October 20th, consisting of 1618 machines up to equivalence, or 3067 individual machines. This means 73 newly solved machines, a 4% reduction.
  • @Bricks shared a machine which they thought could be susceptible to block-analysis based on a method they call Subtape Saturation Heuristic. Shawn Ligocki's analysis, simulated by @Bricks showed the machine to halt with a sigma score of 4,419,340,317.
  • Analysis by Racheline showed a machine to be non-halting.
• BB(7):
  • Andrew Ducharme has continued reducing the number of holdouts with Stage 4 of Phase 2. Initially, in the beginning of the month there were 22,801,601 holdouts, and 22,721,168 holdouts remain. (0.35% reduction)
• BB(4,3):
  • Terry Ligocki has begun phase 2 of holdout reduction, reducing the number of holdouts from 460,916,384 to 9,401,447. (97.96% reduction)
• BB(3,4):
  • XnoobSpeakable and Lúkos are running filters in the domain under Phase 2, reducing the holdout count from 434,787,751 to 15,136,283. (96.6% reduction)
• BB(2,5):
  • Peacemaker II gave an informal proof of a machine never halting, making the informal holdout count 67.
• BB(2,6):
  • Andrew Ducharme has completed stage 3 of phase 2, reducing the number of holdouts from 873,469 to 870,085. (0.39% reduction)

Theory

TODO: Update this section after studying existing literature a bit more.

Linear-Inequality Affine Transformation Automata (LIATA) were introduced as a generalization of the BMO1 rules:

• @Bard proved that 3 dimension LIATA are Turing complete: [1]
• @star proved that 2 dimension LIATA are Turing complete: [2]
• BMO1 is a 2d-LIATA so this provides some sense for the difficulty of the problem.

Deciders

TODO: Add -d's cpp quick_sim project. Discord source: https://discord.com/channels/960643023006490684/1226543091264126976/1426265937455222794

TODO: Add Katelyn's inductive project. Discord source:https://discord.com/channels/960643023006490684/1369339127652159509/1419016459560161280 https://discord.com/channels/960643023006490684/1095740122139480195/1427714010697961534