1RB1LD 1RC1RB 1LC1LA 0RC0RD

1RB1LD_1RC1RB_1LC1LA_0RC0RD (bbch) is the current Blanking Busy Beaver BLB(4,2) and Beeping Busy Beaver BBB(4,2) champion, creating a blank tape after 32,779,477 steps and quasihalting after 32,779,478 steps at which point it becomes a translated cycler with period 1. It was discovered and reported by Nick Drozd in 2021.

In 2026, this machine was proven in Rocq to be the BBB(4) winner when considering beep states.

Analysis by Shawn Ligocki

Let $$D(a,b):=0^{\mathrm{\infty }}{1}^a{0}^b\text{D}\text{<mo>></mo>}{0}^{\mathrm{\infty }}$$then:$$\begin{array}{lccc}D(a+3,& b)& \to & D(a,b+5)\\ D(0,& b)& \to & \text{Blank}\\ D(1,& b)& \to & D(b+2,4)\\ D(2,& b)& \to & D(b+3,4)\\ \end{array}$$let $D(a):=D(a,4)$, then we can simplify to:

$$\begin{array}{lcc}D(3k)& \to & \text{Blank}\\ D(3k+1)& \to & D(5k+6)\\ D(3k+2)& \to & D(5k+7)\\ \end{array}$$Starting from $D(2)$ (at step 19) we get the trajectory:

$$\begin{array}{lllllllllll}D(2)& \to & D(7)& \to & D(16)& \to & D(31)& \to & D(56)& \to & D(97)\\ & \to & D(166)& \to & D(281)& \to & D(472)& \to & D(791)& \to & D(1322)\\ & \to & D(2207)& \to & D(3682)& \to & D(6141)& \to & \text{Blank}\\ \end{array}$$which has the remarkable luck of applying this Collatz-like map 14 times before reaching the blanking config (expected # of applications before halting is 3).

See also, previous analysis in 2021: https://www.sligocki.com/2021/07/17/bb-collatz.html

Relation to other machines

The map and trajectory are equivalent to that of the BB(5) champion. For all $k$, let $f(k)$ be the number such that $D(k)\to D(f(k))$, or $\text{HALT}$ if $D(k)\to \text{Blank}$, and let $g$ be the map simulated by the BB(5) champion. Then:

$$\begin{array}{lcccccc}g(2(3k)+2)& =& g(6k+2)& =& \text{HALT}& =& f(3k)\\ g(2(3k+1)+2)& =& g(6k+4)& =& 10k+14& =& 2f(3k+1)+2\\ g(2(3k+2)+2)& =& g(6k+6)& =& 10k+16& =& 2f(3k+2)+2\\ \end{array}$$

So the size of this machine's BLB output is tied to the size of the BB(5) champion's output.