TMBR: August 2025

This Month in Beaver Research for August 2025.

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

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 Quick_Sim 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
• TODO: BB(7) holdout list reduced from 86,129,304 to 72,470,054.

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.