TMBR: June 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

Busy Beaver for Lambda Calculus:

• BBλ(39) was solved on 10 June, showing that BBλ(39) $=5\cdot 3^{3^{3^3}}+6$.
• Work began on BBλ(40), which was reduced to 2 holdouts^[1] and BBλ(41), which was reduced to [TODO] holdouts.
• A list of BBλ(42) holdouts was also added to the BBλ-spreadsheet.^[2]

SKI Calculus:

• New champions were discovered for Ξ₀(10), Ξ₀(13), Ξ₀(14), Ξ₀(15), Ξ₀(16), Ξ₀(17), Ξ₀(18) and Ξ₀(19). Ξ₀(10) was shown to be larger than Graham's number.
• New SK calculus champions were discovered for Ξ₀_SK(18) and Ξ₀_SK(26), showing Ξ₀_SK(26) to be larger than Graham's number.
• New BCKW system champions were discovered for Ξ₀_BCKW(7), Ξ₀_BCKW(8) and Ξ₀_BCKW(9).

CounterScript:

• On 12 June, sheep constructed a size 84 champion with score $2^{q(5)}$^[3], where q is a fast growing function arising from Laver tables. A day later, sheep showed that BBCS(89 + k) $\ge 2^{P^k(0)}$ where P(x) = q(x) + 1.^[4]
• On 8 June, sheep discovered a new BBCS(25) champion.^[5]
• On 15 June, Shawn Ligocki discovered new champions for BBCS(13)^[6] and BBCS(14).^[7]
• On the same day, Azerty completed the enumeration of BBCS(12), leaving an initial 866 holdouts.^[8] A second enumeration the next day left only 477 holdouts.^[9]
• On June 20, sheep discovered two new champions (first, second) for BBCS(19) in quick succession.

Misc

• TODO: GPU deciders

Holdouts

• BB(6)
  • mxdys solved four TMs with FAR.^{[10][11][12][13]}
  • prurq solved another four TMs with FAR.^[14][15][16]
  • A TM was shown to be non-halting by hipparcos, which was later confirmed in Rocq by mxdys.
• BB(7)
  • Using new GPU based deciders, David SG reduced the number of remaining holdouts from 17,823,260 to 16,949,557, a 4.90% reduction.^[17]
  • Andrew Ducharme further reduced the amount of holdouts by 7.12% to 15,743,264 by running mxdys's FAR decider with various parameters.^[18]
• BB(2,6)
  • Andrew Ducharme reduced the number of holdouts from 413,513 to 412,086, a 0.35% reduction.^[19]
• BB(2,7)
  • Terry Ligocki enumerated 100K more subtasks, increasing the number of holdouts to 1,392,222,673. A total of 450K subtasks out of the 1 million subtasks (or 45%) have been enumerated.