TMBR: September 2026

2026-09-16T17:49:56Z

Seiji: BB(6): Space Needle extended to 100,000,000 terms (independent computation, September 2026) + step-count laws

{{TMBRnav|August 2026|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).''

== BB Adjacent ==

TODO: [[Cyclic Tag]]

== Holdouts ==
* [[BB(6)]]
** By September 4, mxdys solved all 16 Counter Class 1 TMs.[https://discord.com/channels/960643023006490684/1239205785913790465/1545487019218698290]
** On September 4, Discord user fifi_49513 claimed to have proven a TM non-halting. [https://discord.com/channels/960643023006490684/1545543031468462150/1545543031468462150] This machine was later confirmed non-halting in Rocq on September 5 by mxdys. [https://discord.com/channels/960643023006490684/1545543031468462150/1545682001481175095]
** 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). [https://github.com/oohzaim1-afk/space-needle-100M]
* [[BB(2,7)]]
** Terry Ligocki enumerated 20K more subtasks, increasing the number of holdouts to '''2,713,007,927'''. A total of 870K subtasks out of the 1 million subtasks (or '''87%''') have been enumerated.

== Misc ==
* On September 5, [[User:XnoobSpeakable|XnoobSpeakable]] began an opt-in archival project of the bbchallenge Discord server[[Category:This Month in Beaver Research|2026-09]] .[https://discord.com/channels/960643023006490684/960643023530762341/1545742987710038088]

Space Needle

2026-09-16T17:44:05Z

Seiji: Trajectory extended: independent computation to 100,000,000 terms (Sept 2026) — no power of 2; new max v2 = 25 at n = 23,145,881; final value published with SHA-256; step-count laws linked

{{machine|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD}}{{unsolved|Does Space Needle run forever?}}
{{TM|1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD}} is a [[probviously]] non-halting [[BB(6)]] [[Cryptid]] first discovered and analyzed by @mxdys on [https://discord.com/channels/960643023006490684/1239205785913790465/1326911501357023296 9 Jan 2025]. Mxdys derived the lower level rules, and Racheline and Katelyn Doucette independently derived the higher level rules later on.

Determining whether this machine halts requires proving that a highly chaotically growing sequence never intersects with a value equal to <math>2^n</math>. It is believed to be a [[Cryptid]], because proving this seems to require a deep understanding of how addition of two integers affects the number of factors of 2 in the sum, and/or much stronger ways of characterizing when two sequences can share terms in common. Both of which are believed to be out of reach of current mathematics.

This machine is named "Space Needle", and it is the [[Beaver Math Olympiad#Unsolved problems|BMO's 6th problem.]]

== Analysis ==
=== Low Level Rules ===
The original reported low level rules derived by @mxdys:

<pre>
1RB1LA_1LC0RE_1LF1LD_0RB0LA_1RC1RE_---0LD

start: (3,1)
(0,2+c) --> (4+c,1)
(1,c) --> halt
(2+2b,c) --> (7+5b+c,1)
(3+2b,c) --> (b,4+b+c)

(b,c) := 0^inf <A 1^b 00 1^c 0^inf
</pre>

Or the following rewritten form by Andrew Ducharme:

<pre>
start: (3,1)
(1,c) --> halt
(2b,c) --> (2+5b+c,1)
(2b+1,c) --> (b-1,3+b+c)

(b,c) := 0^inf <A 1^b 00 1^c 0^inf
</pre>

=== Higher Level Rules ===
Further derivation and analysis of the higher level rules took place on [https://discord.com/channels/960643023006490684/1400226828396003480 July 31 2025] 
The low level rules can be abstracted up a level by noticing that the c parameter always gets set to a consistent starting value of 1 when the rule triggered by b being even is encountered. It's possible to, through induction, generate a mapping from each <math>(b_n, 1)</math> to the next <math>(b_{n+1}, 1)</math>. This allows one to completely eliminate the c parameter and have a set of rules entirely dependent on the value of b.

[https://discord.com/channels/960643023006490684/1400226828396003480/1400226828396003480 Rules derived by Katelyn Doucette:]
<pre>
Let v(b) denote the largest power of 2 that divides b.

START: b = 6

If b = 2^v(b):
HALT
else:
b --> b + v(b) + 3/2 * (b/2^v(b) - 1)
</pre>

Important to note, the actual machine seems to work with an alternative formulation of the above rules where b and its halting problem are shifted down by 3. The version with b shifted up by 3 is shown instead due to stylistic reasons, as well as improved computation speed.

== Trajectory ==
At a high level, the machine's halting problem is governed by a highly chaotically growing sequence. The first few terms of the sequence are:
<math display="block">\begin{array}{|c|}\hline 6 \rightarrow 10 \rightarrow 17 \rightarrow 41 \rightarrow 101 \rightarrow 251 \rightarrow 626 \rightarrow 1095 \rightarrow 2736 \rightarrow 2995 \rightarrow \cdots\\\hline\end{array}</math>
This sequence has been calculated out to over 17 million terms, with the final value of b calculated before the program was terminated reaching <math> >10^{4,800,000} </math>. No power of 2 was ever encountered (which would lead to halting), and the most factors of 2 any term had was 24.

''Update (September 2026):'' an independent computation (Seiji) extended the trajectory to 100,000,000 terms. No power of 2 was encountered, and the most factors of 2 any term had was 25, first occurring at n = 23,145,881. The final term has 94,125,050 bits (28,334,464 decimal digits). The full final value is published with a SHA-256 anchor, and the trajectory was verified by two independent implementations over the full range (400/400 checkpoints). Artifacts and hashes: https://github.com/oohzaim1-afk/space-needle-100M. The same repository also contains this machine's exact macro-transition step-count laws (L0–L4), with formal proofs for L1 and L3 and machine-certified checkers.

The growth rate of this sequence was characterized by @LegionMammal978 as below: 
Assuming that the lower bits of b are random and uniform (which is well-supported empirically), it grows by an average log-factor of <math>0.652355</math>, which corresponds to <math>O(1.92006^b)</math>, plus or minus a random walk on the log scale.

== See also ==

* {{TM|1RB1LA_0LC0RC_1LE1RD_1RE1RC_1LF0LA_---1LE}}

[[Category:BB(6)]][[Category:Cryptids]]

BusyBeaverWiki - User contributions [en]

TMBR: September 2026

Space Needle