TMBR: October 2025

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This Month in Beaver Research

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A brave busy beaver confronts the dreaded Antihydra. Copyright Nico Roper. Commissioned for Why Busy Beaver Hunters Fear the Antihydra.

This Month in Beaver Research for October 2025.

This month we had an eclectic mix of results. We're continuing to see massive holdout reduction at the peripheral domains with a 98% holdout reduction for BB(4,3) while Polygon identified a new BB(4,3) champion. A new type of Cryptid has been explored: Piecewise Affine Functions. Ben Brubaker wrote-up a personal blog post describing Antihydra to a lay audience. @coda designed some physical disks for computing TM transitions. And @-d is developing a C++ version of Quick_Sim.

Memory map visualization of a Turing machine by @vonhust.

Blog Posts

• 22 Oct 2025. Ben Brubaker. Why Busy Beaver Hunters Fear the Antihydra. (Hacker News thread)

Champions

• Polygon identified a new BB(4,3) champion with a score of over $10{\uparrow }^{4}4$ (1RB1RD1LC_2LB1RB1LC_1RZ1LA1LD_0RB2RA2RD (bbch)). This TM was first proven to halt by Pavel Kropitz in May 2024, but its runtime was not known at the time.
• @Peacemaker II verified the BB(6) champion 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch) (and the other members of its "family") and calculated a more precise sigma score of 10↑↑10↑↑10↑↑8.10237 for it.^[1]
• @zts439 proved that Bug(8,8) = 506.^[1]

Theory

Piecewise Affine Functions (PAF) were explored as a generalization of the BMO1 rules:

• @Bard proved that 3 dimension PAF are Turing complete.^[1]
• @star proved that 2 dimension PAF are Turing complete.^[2][3]
• Shawn Ligocki wrote up a proof sketch that 2-region PAF are Turing complete.^[4]
• It was discovered that Amir Ben-Amram had already proven both of these results in 2015 (both the 2-dim and the 2-region results).
• BMO1 is a 2-dim, 2-region PAF so this provides some sense for the difficulty of the problem.
• This introduces a new type of Cryptids separate from previous Collatz-like ones.

Deciders

• Inductive deciders
  • @-d is developing a C++ version of Quick_Sim. It can currently solve "Diff Rules" (L1 Inductive Rules). It is 6-10x faster than the original python implementation.^[5][6]
  • Katelyn Douchette is working on an automated inductive decider.^[7][8] (see inductive proofs)

Misc

• @coda shared a mechanical implementation of Antihydra^[9] and @zts439 3d-printed a prototype.^[10]

• @Bricks shared a method to estimate susceptibility to Block Analysis and a spreadsheet of BB(6), BB(3,3) and BB(2,5) holdouts quantified by it.^[11][12]
• @Pomme, building onto the work of mxdys, star and Shawn solved BMO #7: 1RB1RF_1RC0RA_1LD1RC_1LE0LE_0RA0LD_0RB--- (bbch)^[1].

Holdouts

BB(6) Holdouts count decrease overtime.

BB Holdout Reduction by Domain

Domain	New Holdout Count	Previous Holdout Count	Holdout Reduction	% Reduction
BB(6)	1618	1691	73	4.3%
BB(7)	20,405,295	22,801,601	2,396,306	10.5%
BB(4,3)	9,401,447	460,916,384	451,514,937	98.0%
BB(3,4)	15,136,283	434,787,751	419,651,468	96.6%
BB(2,6)	870,085	873,469	3384	0.4%

Details

• 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. Quick_Sim confirmed the result.
  • Analysis by Racheline showed a machine to be non-halting.
  • mxdys decided a machine from the 50 Random Holdouts introduced back in August, making 10/50 solved.
  • Peacemaker II mentioned a machine which visits only 24 cells after a million steps. mxdys proved the machine nonhalting.
  • Pomme* solved BMO 7.
• BB(7):
  • Andrew Ducharme has continued reducing the number of holdouts with Stage 4 of Phase 2. Afterwards, Terry Ligocki ran Stage 5 of Phase 2. Initially, in the beginning of the month there were 22,801,601 holdouts, and 20,405,295 holdouts remain. (10.51% reduction)
• BB(4,3):
  • Terry Ligocki finished 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 64.
• 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)