TMBR: April 2026

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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 Month in Beaver Research for April 2026. This month, a new Cryptid was discovered in BB(6) by Discord user sheep, and BMO 8 was added to BMO. Two informally proven machines were formalised into Rocq in BB(2,5), and Katelyn Doucette created a visualizer for Fractran space-time diagrams. We also shot below 18 million holdouts for BB(7).

BB Adjacent

• General Recursive Function
  • 3 Apr: Jacob Mandelson proved the values up to BBµ(7).^[1]
  • 8 Apr: Jacob constructed a size 141 Cryptid.^[2]
  • 12 Apr: Shawn Ligocki enumerated all Primitive Recursive Functions (GRF w/o M) up to size 18, finding two new champions and guaranteeing that anything that beats them would have to use the Min operator.^[3][4]
  • 16 Apr: Shawn built a size 100 GRF that surpasses Graham's number.^[5]
• TODO: BB\ (Busy Beaver for Lambda Calculus)

Misc

• Katelyn Doucette completed a visualizer for Fractran space-time diagrams.^[3]

Holdouts

• BB(6)
  • Discord user sheep discovered^[4][5] a new Cryptid, 1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE (bbch), similar to Space Needle. A classification of Cryptids is now being worked on, where this machine, for example, could belong to a class of Needles (along with Space Needle).
  • BMO 8 was added to the Beaver Math Olympiad: 1RB0LD_0RC1RB_0RD0RA_1LE0RD_1LF---_0LA1LA (bbch)
  • Alistaire simulated a machine to 1e15.
  • Discord user The_Real_Fourious_Banana simulated another TM to 1e15, reducing the 1e14 holdout count to 169 and the 1e15 holdout count to 235.
  • mxdys released a new holdouts list of 1119 machines, the reduction mostly (except for one TM) came from new equivalences.
  • Later, mxdys released a new holdouts list of 1104 machines where more equivalence classes have been merged.
  • TODO: Add BB6 holdouts decrease graph in 2026: https://discord.com/channels/960643023006490684/1239205785913790465/1492615938824999034
• BB(7)
  • Further filtering by Andrew Ducharme reduced the number of holdouts from 18,036,852 to 17,823,260.^[6] (A 1.18% reduction)
• BB(4,3):
  • In phase 2 stage 3, Andrew Ducharme reduced the number of holdouts from 9,401,447 to 5,641,006, a 40.00% reduction.^TODO
• BB(3,4):
  • Andrew ducharme reduced the holdout count from 12,435,284 to 12,049,358 (a 3.10% reduction) with mxdys's FAR decider.
• BB(2,5):
  • On 1 April 2026, Discord user mammillaria shared a Lean formalisation of the BMO 3 problem and its solution, which he created using Aristotle AI. Then mxdys formalised the result in Rocq using LLMs, reducing the formal holdout count to 67, still with 60 informal holdouts.
  • On 2 April 2026, mxdys solved BMO 3 variant 1RB0RA3LA4LA2RA_2LB3LA---4RA3RB (bbch) using an LLM, reducing the formal holdout count to 66. The proofs for BMO 3 and its variant are available at https://github.com/ccz181078/busycoq/blob/BB6/verify/BMO3.v.
  • 1RB2RA3LA4LA2RB_2LA---1LA1RA3RA (bbch) and 1RB3LA4LA2RB1LA_2LA4RB---3RA3LA (bbch) were simulated until halting by prurq using Quick_Sim.^[TODO][TODO]
• BB(2,6)
  • Andrew Ducharme reduced the number of holdouts from 545,005 to 536,112 via Enumerate.py, a 1.63% reduction.^{[TODO][TODO][TODO]}
• BB(2,7)
  • Terry Ligocki enumerated 110K more subtasks, increasing the number of holdouts to 660,441,047. A total of 210K subtasks out of the 1 million subtasks (or 21%) have been enumerated.^[7]