TMBR: May 2026

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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).

BB Adjacent

General Recursive Functions:

• Some new champions were found:
  • BBµ(16) ≥ 47, found by Shawn on 5 May: M(C(R(S, R(P(2,1), R(P(3,2), C(R(S, P(3,1)), P(5,3), P(5,2))))), P(1,1), S)).^[1]
  • BBµ(17) ≥ 2090, found by Shawn on 8 May: M(C(R(P(1,1),R(P(2,1),R(R(R(S,C(S,P(3,2))),P(4,1)),P(5,2)))),P(1,1),S)).^[2]
  • BBµ(93) ≥ Graham's number, designed by racheline on 19 May using a binary encoding of weak goodstein sequences.^[3][4]
• Some new cryptids were hand-built:
  • Size 56, by Shawn on 2 May (simulating 5x+1 problem starting at 7).^[5]
  • Size 49, by aparker, star and Shawn on 3 May (simulating Brocard's problem).^[6]
• The first non-trivial divergent GRF was found (size 15). It halts iff there exists some n ≥ 1 such that n+3 divides $Tetr(n)=\frac{n(n+1)(n+2)}{6}$.^[7] aparker^[8] and star^[9] proved that there is no such n.
• On 5 May, Shawn implemented a size 133 GRF that halts iff there was a counter-example to Fermat's Last Theorem: fermat.mgrf.^[10]
• On 6 May, Shawn found a chaotic (potentially Cryptid) size 15 GRF: C(R(S, R(C(S, P(2,1)), C(R(P(1,1), P(3,1)), P(4,2), P(4,1)))), P(1,1), S).^[11]
• All M(PRF) GRFs were decided up to size 12 with max score 5.^[12]
• All PRFs were decided up to size 22 with max score $\approx 2^{2^{230}}$. All except 11 were decided by direct simulation, but 11 overflowed u64 and needed to be evaluated by more clever programs.^[13]

SKI Calculus:

• Many new champions were discovered for the Busy Beaver variants of SKI calculus, SK calculus and BCKW system.

CounterScript:

• On 1 May, discord user sheep discovered a BBCS(65) champion beating Graham's number.^[14] On 3 May, sheep improved this champion to size 64.^[15]
• BBCS(10) was shown to be exactly 16.^[16]
• On 6 May, sheep discovered a new champion for BBCS(21)^[17] and on 11 May a new champion for BBCS(25).^[18]
• On 16 May, Azerty enumerated BBCS(11). [TODO: bug found, may have to be removed]^[19]

Fractran:

• On 28 May, @-d began enumerating BBf(23).^[20] The enumeration was completed by 31 May, leaving an initial 790,335 holdouts, which where reduced to 29,250 holdouts.^[21]

TODO: Check for other BB-like functions' progress

Misc

• mxdys made his inductive decider for history-linear machines compilable into an executable file.^[22]
• TODO: BB Bignum Bakeoff
• TODO: GPU deciders
• LegionMammal shared exploration they have made into searching for alternative "Axiom of Infinity" like statements and which ones imply which others.^[23]

Holdouts

BB Holdout Reduction by Domain

Domain	Previous Holdout Count	New Holdout Count	Holdout Reduction	% Reduction
BB(4,3)	5,641,006	5,127,263	513,743	9.11%
BB(3,4)	12,049,358	11,362,197	687,161	5.70%
BB(2,5)	66	65	1	1.52%
BB(2,6)	536,112	413,513	122,599	22.87%

• BB(2,5)
  • Andrew Ducharme found a FAR solution for a machine, which previously had an informal nonhalting argument. On June 1st, mxdys provided the Rocq proof, reducing the Rocq-holdout count to 65.
• BB(4,3)
  • Andrew Ducharme reduced the number of holdouts from 5,641,006 to 5,127,263, a 9.11% reduction, with mxdys's new inductive decider.^[24]
• BB(3,4)
  • XnoobSpeakable completed Stage 2 of Phase 3, reducing to number of holdouts from 12,049,358 to 11,362,197, a 5.70% reduction.
• BB(2,6)
  • Andrew Ducharme reduced the number of holdouts from 536,112 to 527,232 via Enumerate.py and TM-enum, a 1.66% reduction.^[25][26]
  • Using the new inductive decider by mxdys and TM-enum, Andrew Ducharme first reduced the number of holdouts from 527,232 to 501,914^[27][28], then to 439,120 again using the inductive decider.^[29] In total this was a 16.71% reduction.
  • Using FAR, Andrew Ducharme further reduced the number of holdouts from 439,120 to 418,127, a 4.78% reduction.
  • Using some mxdys deciders, Andrew Ducharme continued reducing the number of holdouts through May 17, from 418,127 to 413,513 (a 1.10% reduction).^{[13][14][15][16]}
• BB(2,7)
  • Terry Ligocki enumerated 130K more subtasks, increasing the number of holdouts to 1,087,732,936. A total of 350K subtasks out of the 1 million subtasks (or 35%) have been enumerated.^[17]