TMBR: August 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).

BB Adjacent

• TT (Terminating Turmites)
  • On August 2, Discord user creeperman7002 discovered a new step and score champion for TT(3,3), 1PB2PA0PC_2TB1PC2PB_2PB1PZ0PA, running for 715,764,479 steps.^[1]
  • On August 8, Discord user creeperman7002 discovered a new step champion for TT(3,3), 1PB0TC1PA_2PC1PZ2PA_2TC0PA2PC, running for 2.27 × 10²⁴⁸ steps.^[2]

• Pebble Automaton
  • On August 8, AlephSquirrel introduced the Busy Beaver function for Pebble Automata, peBBle(n), and proved the values of peBBle(n) for sizes 1 to 3.^[3]
  • On the same day, sheep discovered an infinite family of automata, including the champion for peBBle(4).^[4]
  • On August 9, AlephSquirrel discovered a new champion for peBBle(5).^[5]
  • On August 10, Deccy discovered a new champion for peBBle(6)^[6]
  • On August 11, Deccy discovered a new champion for peBBle(7).^[7]
  • On August 13, mathmasterzach discovered 3 new peBBle(8) champions in quick succession,^[8][9][10] and the previous peBBle(9) champion, which ran for 323 steps.^[11]
  • On August 16, AlephSquirrel proved the peBBle(4) champion to be optimal.^[12]
  • On August 17, AlephSquirrel tweaked the peBBle(9) champion to run for 63577 steps.^[13]

TODO: Wang Tiles

TODO: SKI Calculus

TODO: Cyclic Tag

Holdouts

• BB(6)
  • On August 1, mxdys added BMO 9 and BMO 10 to the Beaver Math Olympiad.^[14][15]
  • On August 3, prurq found three machines to halt.^[16][17][18]
  • On August 5, q64 simulated another TM to halt.
  • On August 15, Discord user jacobrschwartz proved another TM halting.
  • Later that day, mxdys released a new holdouts list of 1,016 machines up to equivalence, this is a 4.51% reduction compared to the previous holdouts lists.
  • On August 24, Discord user flintt_6 informally proved another TM halting.
  • On August 26, q64 proved another TM halting.
  • On August 30, mxdys released another holdouts list of 1,003 machines up to equivalence.
  • On August 31, mxdys released another holdouts list of 999 machines up to equivalence.
  • TODO: BMO 9 and BMO 10
• BB(2,7)
  • Terry Ligocki enumerated 150K more subtasks, increasing the number of holdouts to 2,651,060,579. A total of 850K subtasks out of the 1 million subtasks (or 85%) have been enumerated.