TMBR: September 2026

Prev: August 2026

This Month in Beaver Research

Next: October 2026

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).

Blog Posts

• September 1, 2026. Nick Drozd. A Tetrational Self-Cleaning Turing Machine.

Champions

• On September 1, nickdrozd discovered a new BLB(3,3) champion which runs for over 10↑↑6.54 steps before blanking the tape.^[1] Note that this is not a new BBB(3,3) champion.
• On September 9, @-d discovered a new BBf(24) champion which runs for over 9.000 × 10^14,448 steps.^[2]

BB Adjacent

• Cyclic Tag
  • On September 1, Azerty enumerated all BBCT programs up to size 14, proving the values of BBCT(n) for n < 7.^[3]
  • On September 2, sheep discovered new champions for BBCT(11),^[4] BBCT(13),^[5] and BBCT(14).^[6]
  • On September 5, Azerty proved BBCT(7) = 13.^[7]

• Cellular Automata Busy Beaver
  • On September 1, Azerty introduced an extension to mxdys's BB1dCA(n) function, BBCA(n),^[8] and discovered a champion for BBCA(3).^[9]
  • On September 2, Azerty discovered champions for BBCA(2), BBCA(3), BBCA(4), BBCA(2,3), and BBCA(2,4).^[10]
  • On September 20, Azerty discovered new champions for BBCA(4), BBCA(3,3), and BBCA(2,4), and proved BBCA(2,3) = 4.^[11]

• General Recursive Function
  • On September 5, after Shawn Ligocki renewed interest in a certain chaotic BBµ(16) holdout last month, sheep devised a method to accelerate the simulation of the holdout.^[12] This allowed her to confirm that, if the holdout returned a value, it would be larger than 3.124 × 10¹⁶.^[13]
  • On September 7, the aforementioned method was ported to Rust by @ZTS439, allowing them to extend the bound to 4.999 × 10¹⁷.^[14]

• PLUSMINUS^[15]
  • On September 8, @Jack introduced PLUSMINUS, a new esoteric language.^[16] Though its completeness is unknown, this language is not expected to be Turing-complete.^[17]
  • On September 9, @mathmasterzach discovered champion PLUSMINUS strings for sizes 9,^[18] 11,^[19] 12,^[20], 15, and 16.^[21]
  • On the same day, @Furious discovered champion PLUSMINUS strings for sizes 1 to 8^[22] and size 10-14.^[23]
  • On the same day, racheline provided an estimate for the halting times of PLUSMINUS programs.^[24]
  • On September 10, @Furious converted the rules of PLUSMINUS into a piecewise mathematical function.^[25]
  • On September 11, @Jack discovered champion PLUSMINUS strings for sizes 20, 23, 32, 36, and 40.^[26]

• Post Tag System (alternate definition)
  • On September 12, Azerty introduced an alternate formulation of the Busy Beaver function for Post Tag Systems, where a system in the BB_PT*(m, n) domain may continue running even if it reduces to less than m symbols;^[27] in this formulation, the BB_PT(11) cryptid 11111_20_1 is non-halting.
  • On September 16, Azerty enumerated all 2-tag systems with size less than or equal to 9^[28], and proved the values of BB_PT*(n) for sizes 1 to 5.
  • On the same day, sheep discovered a new BB_PT*(9) champion.^[29]
  • On September 30, sheep discovered a 2-tag size 11 cryptid, showing BB_PT*(9) is hard;^[30] Azerty expects there to be smaller cryptids,^[31] likely referring to systems with their tape length history similar to a biased random walk.
• Fractran
  • On September 13, @-d completed enumeration of BBf(24), leaving an initial 6,733,785 holdouts, which were reduced to 231,432 holdouts, or 10,838 holdouts if ai-created deciders are used.^[32]

• Sequential Substitution System
  • On September 16, @andrewthebonfire and Azerty introduced the Busy Beaver function on Sequential Substitution Systems,^[33][34][35] with Azerty later finding BBSS(n) champions for sizes 1 to 8, and proving the values of BBSS(n) for sizes 1 to 4.^[36]
  • On September 17, @andrewthebonfire provided a method to convert Turing machines into sequential substitution systems.^[37] He also showed that sequential substitution systems with a binary alphabet (i.e. two symbols) are still Turing-complete.^[38]
  • On September 30, sheep discovered a size 17 cryptid, showing BBSS(17) is hard.^[39]

• Register Machine
  • On September 18, @mineralli discovered a new MBB(9) champion.^[40] sheep then discovered four new champions for MBB(9) in the four following days.^[41]
  • On the same day, @elgan65536 discovered a new MBB(8,2) champion.^[42]
  • On September 25, sheep discovered an MBB(10) champion,^[43] which was shown to halt in around 10^2,629,437 steps by Shawn Ligocki.^[44] This champion prompted a rework of Ligocki's hydra-like rule accelerator.^[45]
  • On September 29, @mineralli discovered a 5-register, 19-state machine which halts iff the Erdős ternary conjecture is false.
  • On the same day, sheep discovered a 7-register, 28-state machine beating Graham's number.^[46]
  • On September 30, @mineralli discovered a new MBB(11,2) champion,^[47] improved upon by sheep.^[48]
• Pebble Eating
  • On September 30, sheep discovered an 86 pebble cryptid, showing peBBle(86) is hard.^[49]

Holdouts

• BB(6)
  • By September 4, mxdys solved all 16 Counter Class 1 TMs.^[50]
  • On September 4, Discord user fifi_49513 claimed to have proven a TM non-halting. ^[51] This machine was later confirmed non-halting in Rocq on September 5 by mxdys. ^[52]
  • On September 8, mxdys released a new holdouts list of 964 machines up to equivalence.^[53]
  • On September 14, an independent computation by Seiji extended the Space Needle trajectory to 100,000,000 terms: no power of 2 was ever encountered, and the most factors of 2 any term had was 25 (first occurring at n = 23,145,881; the previously documented state was >17,000,000 terms with max 24). The full final value (94,125,050 bits) is published with SHA-256 anchors, together with 400 checkpoints cross-verified by two independent implementations, and exact macro step-count laws of the machine (L0–L4; L1 and L3 formally proven). ^[54]
  • On September 16, mxdys released another holdouts list of 855 machines up to equivalence.^[55]
  • On September 24, mxdys released another holdouts list of 815 machines up to equivalence.^[56] The newly solved holdouts include:
    • BMO9
    • Two shift-overflow bell counters
    • 31 shift-overflow mixed-digits counters with similar behavior
    • two new translated cyclers (also shift-overflow mixed-digits counters) with high period
    • two other shift-overflow counters
  • On September 24, Discord user q64 simulated a TM to halt^[57]
  • The halting status of 1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB--- (bbch) was formalized without new axioms by avoiding the use of Baker–Wüstholz theorem.
  • On September 30, mxdys released another holdouts list of 797 machines up to equivalence.^[58]
• BB(2,5)
  • On September 8, mxdys released a new holdouts list of 60 machines.^[59]
  • On September 24, prurq used Aristotle AI and GPT6 to solve 20 of the 60 remaining holdouts.^[60]
• BB(2,7)
  • Terry Ligocki enumerated 31K more subtasks, increasing the number of holdouts to 3,062,427,379. A total of 980K subtasks out of the 1 million subtasks (or 98%) have been enumerated.

Misc

• On September 2, RobinCodes began work on updating the BB(6) holdout spreadsheet.^[61]
• On September 5, XnoobSpeakable began an opt-in archival project of the bbchallenge Discord server.^[62]
• On September 23, Joseph M. Shunia published a preprint presenting a 120-state TM that halts iff Riemann hypothesis is false.^[63]
• On September 22, Twitter user @_Mira___Mira_ claimed to have discovered a 387-state TM whose halting status is independent of ZFC with GPT-6 Astra.^[64]
  • On September 24, they claimed to have found a 314-state ZFC-independent TM, and a 289-state PA-independent TM.^[65]
  • On September 30, after a series of constant improvements, they claimed to have found a 256-state ZFC-independent TM, and a 238-state PA-independent TM. A Metamath formalization is in progress.^[66]
• On September 28, Twitter user @_Mira___Mira_ discovered a fractal related to the Antihydra conjecture.^[67]