TMBR: August 2025

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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.