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 the Beaver Math Olympiad. Two informally proven machines were formalised into Rocq in BB(2,5). There was a 40% reduction in BB(4,3), and we also passed below 18 million holdouts for BB(7). There's been a lot of discoveries in the Fractran, General Recursive Function, and Lambda Calculus versions of Busy Beaver. Katelyn Doucette created a visualizer for Fractran space-time diagrams. BBf(22) has been solved except for the Fenrir family of Cryptids. The first BBµ champion was found that takes advantage of the minimization (M) operator. Both BBf(100) and BBµ(100) were proven to surpass Graham's number. BBλ(38) was solved and a 74-bit BBλ Cryptid was found.
Racheline created a series of fast-growing programs: a tetrational program of size 29,[2] programs starting from size 86,[3] and programs from size 95, meaning Graham's number fits under size 100. She predicts that one probably exists under size 40, and that it shouldn't be hard to reduce it to at least 60.
Jacob Mandelson proved the values up to BBµ(7) on 3 Apr.[4]
A couple of Cryptids were hand-constructed: size 141, by Jacob on 8 Apr,[5] and size 81, by Shawn Ligocki on 28 Apr.[6]
Shawn built an "Ackermann worm" function with growth of size 83 on 16 Apr and used to it show BBµ(100) > Graham's number.[7]
Jacob extended the Ackermann worm to find a growth function of size 204 on 23 Apr.[8][9]
Shawn enumerated all Primitive Recursive Functions (GRF w/o Min) up to size 20.[10][11][12]
Shawn found a series of new chaotic size 14 champions using the Min operator on 29 Apr, proving BBµ(14) ≥ 32.[13] The longest running takes ~30k sim steps and all size 14 GRF of the form M(PRF) have been simulated out to 10M sim steps.[14]
Shawn is working on a distributed computation version of GRF enumeration so that others can contribute compute.[15]
Jumping Busy Beaver has been introduced, JBB(2,2,n) is known for n = 0 to n = 10, along with some lower bounds on small domains, see the Discord thread.
Misc
ZTS439 explored some properties of summations over the Hydra function .[16]
The Turing Machine 1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB--- has been informally solved for months now. The formal solution depends on a number theory result which would be a major project in of itself to formalise. Therefore, the following statement was formalised: assuming the Baker–Wüstholz core bound for linear forms in logarithms over ℚ, the Turing machine never halts. See Github, Axiom minimal version: Discord, The machine's Discord thread: Link. Note that the formal proofs were made with the help of Claude Opus and Aristotle AI.
mxdys released a new holdouts list of 1119 machines, the reduction mostly (except for one TM, the other informal holdout) came from finding new equivalences. This means there is now only 1 holdout (see above) whose solution has not been fully formalised.
Later, mxdys released a new holdouts list of 1104 machines where more equivalence classes have been merged.
These equivalences were found with the help of -d, see (Discord 1, 2, 3). Equivalences seem to be amongst the last low-ish hanging fruits, with -d estimating about 100-200 equivalences left.
Alistaire and Discord user @The_Real_Fourious_Banana each simulated a TM to 1e15 steps. Combined with the recent equivalence reductions (10 machines total), the number of machines to simulate to 1e14 and 1e15 steps is 161 & 225 respectively.
In phase 2 stage 3, Andrew Ducharme reduced the number of holdouts from 9,401,447 to 5,641,006, a 40.00% reduction. He also found several new high-scoring halters, current places 4 through 8 in the 4x3 Busy Beaver game. 4th place is 1RB1LD2LA_0RC1RZ0RA_1LD2LA0LC_2RD2RC0LD (bbch) with approximate sigma score ~10↑↑1023.47221. [20]
1RB2RA3LA4LA2RB_2LA---1LA1RA3RA (bbch) and 1RB3LA4LA2RB1LA_2LA4RB---3RA3LA (bbch) were simulated until halting by prurq using Quick_Sim.[21][22] These TMs, in addition to 1RB3LA4LA2RB1LA_2LA4RB---3RA3LA (bbch), were shown to halt in 2024 June (see Discord), but step counts and scores for these machines were unknown.
Terry Ligocki enumerated 120K more subtasks, increasing the number of holdouts to 687,123,946. A total of 220K subtasks out of the 1 million subtasks (or 22%) have been enumerated. (see Google Drive) [26][27]
