Antihydra

Antihydra is the 6-state 2-symbol machine https://bbchallenge.org/1RB1RA_0LC1LE_1LD1LC_1LA0LB_1LF1RE_---0RA.

This machine was the first identified BB(6) Collatz-like Cryptid, and is closely related to Hydra.

It simulates the Collatz-like iteration

$${\begin{array}{l}A(2a,&b)&\to &A(3a,&b+2)\\A(2a+1,&b)&\to &A(3a+1,&b-1)&{\text{if}}&b>0\\A(2a+1,&0)&\to &{\text{HALT}}\end{array}}$$

starting from A(8, 0),
using configurations of the form $A(a+4,b)=0^{\infty }\;1^{b}\;0\;1^{a}\;E>\;0^{\infty }$

It was discovered by mxdys on 28 Jun 2024 and shared on Discord [1].

Racheline found that compared to the Hydra iteration, this one starts at (8, 0) rather than (3, 0), and the roles of odd and even a are exchanged (in terms of which increases b by two, and which decrements b or halts). Obstacles to proving the long-run behavior are equally serious. Like the Hydra iteration, this one is biased toward increasing the value of b (assuming equal chances of adding +2 or -1).

There is no halt in the first 11.8 million iterations, by which point b has reached 5890334 (which means that it also does not halt in the first 17690334 iterations) [2].