TMBR: December 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).

This Year in Beaver Research _{(TYBR - "Thank You Beaver Researchers!")}

Holdouts Reductions.

• BB(6) - Reduced from 3571 to 1416 holdouts. Hence, 2155 machines were solved this year. This is a 60% reduction.
• BB(7) - Enumeration was completed, the number of holdouts was reduced from an initial 85,853,789 to 20,405,295 machines, a 76.23% reduction.
• BB(4,3) - Reduced from 460,916,384 to 9,401,447 holdouts, a 97.96% reduction.
• BB(3,4) - Reduced from 434,787,751 to 15,136,283 holdouts, a 96.52% reduction.
• BB(2,5) - Reduced from 217 to 75, a 65.43% reduction.
• BB(2,7) - Enumeration started, 50K of the 1M subtasks have been enumerated (5%).

Champions.

• BB(6) - On 16 June 2025, mxdys discovered 1RB1LC_1LA1RE_0RD0LA_1RZ1LB_1LD0RF_0RD1RB (bbch), running for 10 ↑↑ 11010000 steps. This was surpassed on 25 June when mxdys discovered 1RB1RA_1RC1RZ_1LD0RF_1RA0LE_0LD1RC_1RA0RE (bbch), a TM which runs for $10\uparrow \uparrow 10\uparrow \uparrow 10\uparrow \uparrow 8.10237$ steps.
• BB(7) - Within three days of the start of the enumeration of BB(7), three champions were discovered. The first two were discovered by Shawn Ligocki: 1RB0RF_1LC0RE_1RD1LB_1LA1LD_0RA0LE_1RG0LB_1RZ1RB (bbch) with a sigma score of about 10 ↑↑ 22 and 1RB1RA_1RC0LC_0LD1LG_1LF0LE_1RZ1LF_0LA1LD_1RA1LC (bbch) with a sigma score of about 10 ↑↑ 35. This was followed by the discovery of 1RB0LG_1RC0RF_1LD1RZ_1LF0LE_1RA1LD_1LG1RE_0LB0LB (bbch), achieving a sigma score of about 10 ↑↑ 46, by Terry Ligocki. On 10 May 2025, Pavel Kropitz discovered 1RB0RA_1LC1LF_1RD0LB_1RA1LE_1RZ0LC_1RG1LD_0RG0RF (bbch), a TM which runs for over $2{\uparrow }^{11}2{\uparrow }^{11}3$ steps.

TODO

New Methods.

TODO

Meta.

TODO

BB Adjacent.

• Instruction-Limited Busy Beaver was introduced and calculated up to BBi(7).
• Reversible Turing Machine Busy Beaver values were calculated up to BB_rev(5).
• Terminating Turmites (Relative Movement Turing Machines) were introduced.
• John Tromp introduced the $BB{\lambda }_1(n)$ function for Busy Beaver for lambda calculus with an oracle and computed it up to $BB{\lambda }_1(22)$.
• Instruction-Limited Greedy Busy Beaver gBBi(n) and an Instruction-Limited variant of the Blanking Busy Beaver (BLBi(n)) were introduced. gBBi(n) was computed up to n = 13 and BLBi(n) was computed up to n = 7.
• @savask shared the Bug Game (and fast-growing $Bug(H,W)$ function).
• Busy Beaver for Fractan (BBf) was introduced on 1 Nov by Jason Yuen.^[1] Exact values have been proven up to BBf(19) = 370 and exhaustive enumeration has been run up to size 21 (with BBf(21) ≥ 31,957,632 and 587 holdouts).
• Cyclic Tree Busy Beaver (CTBB) was introduced by @Jack on 14 Nov.^[2] The exact value is known for CTBB(2) = 5 and lower bounds have been found up to size 7 with CTBB(7) > 4↑↑↑↑(4↑↑↑3).

TODO

BB Adjacent

TODO. Register machines, General Recursive Functions.

Holdouts

• BB(6):
  • There are 14 holdouts left to simulate up to 1e12 steps, and 312 to simulate up to 1e13 steps^[1]. The two lists can be found here.
• BB(3,4):
  • TODO: Phase 2 stage 8.
• BB(2,7):
  • Terry Ligocki enumerated 10K more subtasks, increasing the number of holdouts to 150,662,006 and making 50K of the 1 million subtasks enumerated or 5%.