TMBR: September 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

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

• 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.^[5] This allowed her to confirm that, if the holdout returned a value, it would be larger than 3.124 × 10¹⁶.^[6]
  • On September 7, the aforementioned method was ported to Rust by @ZTS439, allowing them to extend the bound to 4.999 × 10¹⁷.^[7]

TODO: PLUSMINUS

• Fractran
  • On September 9, @-d discovered a new BBf(24) champion which runs for over 9.000 × 10^14,448 steps.^[8]
  • 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.^[9]

TODO: Post Tag System (alternate definition)

• Register Machine
  • On September 18, @mineralli discovered a new MBB(9) champion.^[10] sheep then discovered four new champions for MBB(9) in the four following days_.^[11]

Holdouts

• BB(6)
  • By September 4, mxdys solved all 16 Counter Class 1 TMs.^[12]
  • On September 4, Discord user fifi_49513 claimed to have proven a TM non-halting. ^[13] This machine was later confirmed non-halting in Rocq on September 5 by mxdys. ^[14]
  • On September 8, mxdys released a new holdouts list of 964 machines up to equivalence.^[15]
  • 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). ^[16]
  • On September 16, mxdys released another holdouts list of 855 machines up to equivalence.^[17]
  • On September 24, mxdys released another holdouts list of 815 machines up to equivalence.^[18] 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^[19]
  • The halting status of 1RB1LA_1RC1RE_1LD0RB_1LA0LC_0RF0RD_0RB--- (bbch) was formalized without new axioms by avoiding the use of Baker–Wüstholz theorem.
• BB(2,7)
  • Terry Ligocki enumerated 28K more subtasks, increasing the number of holdouts to 2,964,345,750. A total of 950K subtasks out of the 1 million subtasks (or 95%) have been enumerated.

Misc

• On September 5, XnoobSpeakable began an opt-in archival project of the bbchallenge Discord server.^[20]
• On September 23, Joseph M. Shunia published a preprint presenting a 120-state TM that halts iff Riemann hypothesis is false.[21]
• Twitter user @_Mira___Mira_ claimed to have found TKTK-state TM <TKTK update when final TM is published> whose halting status is independent of ZFC with GPT-6 Astra.