A Collatz-like mapping based on modulo 4: do all numbers loop or diverge?

I have designed a new variant of the Collatz conjecture based on modulo 4 remainders, and I am looking for computational data or heuristic analysis regarding its convergence.

Definition of the Mapping

Let $n$ be a positive integer. We define the transition function $g(n)$ as follows based on $n \pmod 4$:

$$g(n) = \begin{cases} \frac{n}{4} & \text{if } n \equiv 0 \pmod 4 \\ 3n + 1 & \text{if } n \equiv 1 \pmod 4 \\ \frac{n}{2} & \text{if } n \equiv 2 \pmod 4 \\ 3n - 1 & \text{if } n \equiv 3 \pmod 4 \end{cases}$$

Observed Behavior & Trivial Cycle

For small values, most numbers seem to quickly hit the trivial cycle: $$\mathbf{4 \to 1 \to 4 \to 1 \dots}$$

For example, starting with $n = 5$:

• $5 \equiv 1 \pmod 4 \implies g(5) = 3(5)+1 = 16$
• $16 \equiv 0 \pmod 4 \implies g(16) = 16/4 = 4$
• $4 \equiv 0 \pmod 4 \implies g(4) = 4/4 = 1$
• $1 \equiv 1 \pmod 4 \implies g(1) = 3(1)+1 = 4$

A Fascinating Example: The Mersenne Number $M_{13} = 8191$

I specifically tested the Mersenne number $8191$ (which is $\equiv 3 \pmod 4$). It undergoes a wild sequence of expansion and contraction before surprisingly collapsing back to 1 in 30 steps: $8191 \to 24572 \to 6143 \to 18428 \to 4607 \to 13820 \to 3455 \to 10364 \to 2591 \to 7772 \to 1943 \to 5828 \to 1457 \to 4372 \to 1093 \to 3280 \to 820 \to 205 \to 616 \to 154 \to 77 \to 232 \to 58 \to 29 \to 88 \to 22 \to 11 \to 32 \to 8 \to 2 \to 1$.

Heuristic Concern

Heuristically, when $n \equiv 1 \pmod 4$, $3n+1$ is always $\equiv 0 \pmod 2$. If it lands on $\equiv 2 \pmod 4$, it is only divided by 2, yielding an overall growth multiplier of roughly $\frac{3}{2}$. If it lands on $\equiv 0 \pmod 4$, it gets divided by 4, yielding a multiplier of $\frac{3}{4}$. Because of the $\frac{3}{2}$ branches, I suspect that some numbers might escape to infinity.

My Questions

1. Has this exact $\pmod 4$ mapping been studied before? If so, what is it called?
2. Does anyone have a script (Python/C++) that can check up to $10^9$ or higher to find the first counterexample that either diverges to infinity or falls into a non-trivial cycle other than $\{1, 4\}$?

I would appreciate any computer search results or probabilistic proofs regarding the behavior of this system!