TMBR: August 2025

This Month in Beaver Research for August 2025, featuring plenty of holdouts reduction in numerous domains, alongside an upgraded TM-visualizer and a couple new BB adjacent games.

Cryptids

• A fast algorithm for Consistent Collatz simulation was re-discovered and popularized. Using it:
  • apgroucher simulated Antihydra to $2^{38}$ iterations. This is actually a result from one year ago, but was rediscovered and added to the wiki. https://discord.com/channels/960643023006490684/1026577255754903572/1271528180246773883
  • Shawn Ligocki simulated 1RB1RA_0RC1RC_1LD0LF_0LE1LE_1RA0LB_---0LC (bbch) out to one additional Collatz reset, demonstrating that (if they halt, which they probviously should) they will have sigma scores $>10^{10^{10^{7}}}$.

Holdouts

• BB(6) holdouts:
  • XnoobSpeakable found 9 new halting TMs in the high exponential runtime range (~10^100000) by running Enumerate.py out to extremely high parameters. https://discord.com/channels/960643023006490684/1239205785913790465/1401470301467836556
  • Andrew Ducharme found a couple surprisingly short running halting TMs in the BB(6) holdouts list with runtime ~10^78. https://discord.com/channels/960643023006490684/1239205785913790465/1407754434523693179
• After the enumeration of BB(7) was completed, Andrew Ducharme ran several deciders on the holdouts list, filtering the original 86,129,304 holdouts down to 60,765,943 in 8 days. https://drive.google.com/drive/u/0/folders/17U0BRpJHTMLtB0poBlOSZhGGp4FkCHIO
• BB(3,3): 9 holdouts were proven non-halting in Rocq (previously known as Coq) by mxdys. 10 holdouts remain, 4 of them solved with moderate rigor. https://discord.com/channels/960643023006490684/1259770474897080380/1410308974275985428

BB Adjacent

• John Tromp introduced the $BB\lambda _{1}(n)$ function for Busy Beaver for lambda calculus with an oracle and computed it up to $BB\lambda _{1}(22)$.
• Instruction-Limited Greedy Busy Beaver gBBi(n) and an Instruction-Limited variant of the Blanking Busy Beaver (BLBi(n)) were introduced. gBBi(n) was computed up to n = 13 and BLBi(n) was computed up to n = 7.

Conference poster for the 31st International Conference on DNA Computing and Molecular Programming, made by Tristan Stérin (cosmo)

Misc

• Iijil shared an algorithm for converting an arbitrary n-state m-symbol TM into a 2-state TM with 3(n+1)m symbols. https://gist.github.com/Iijil1/0d611dbf0a9d52984f72cb14e66a4b28
• Carl K updated his TM web-visualizer to support multi-symbol machines. https://carlkcarlk.github.io/busy_beaver_blaze/v0.2.6/index.html He also extended his series of videos showing TM simulation accompanied by classical music out to some multi-symbol TMs:
  • Bigfoot: https://youtu.be/YvOHWbQNMoY
  • Brady's Surprise in a Box: https://youtu.be/vIG2CvJShRc
  • BB(2,5) champ: https://youtu.be/QpYBzYDdLEY

In the News

• 22 Aug 2025. Ben Brubaker. Quanta Magazine. Busy Beaver Hunters Reach Numbers That Overwhelm Ordinary Math.

Interesting TMs

A collection of interesting TMs that were mentioned on Discord, most because of their space-time diagrams or general behavior.

• 1RB0LE_1RC0RF_1RD---_0LA1RB_1RB1LE_1LD1RF (bbch): Wavy Machine