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Function to calculate the adjacent element distance

I'm exploring a function that takes a non-negative integer vector $$ \theta\in {\mathbb{N}_0}^4$$ and returns the modulus of each adjacent vector components, wrapping around:

$$
s:{\mathbb{N}_0}^4 \rightarrow {\mathbb{N}_0}^4 \\
$$

$$
\begin{align*}
& s(\boldsymbol{\theta}) =
s\left(\begin{bmatrix} x \\ y \\ z \\ w \end{bmatrix} \right) =
\begin{bmatrix}
|y-x| \\
|z-y| \\
|w-z| \\
|x-w|
\end{bmatrix}
\end{align*}
$$

Example:

$$
\begin{align*}
& s\left(\begin{bmatrix} 6 \\ 2 \\ 15 \\ 7 \end{bmatrix} \right) =
\begin{bmatrix}
|2-6| \\
|15-2| \\
|7-15| \\
|6-7|
\end{bmatrix} =

\begin{bmatrix}
|-4| \\
|+13| \\
|-8| \\
|-1|
\end{bmatrix} =

\begin{bmatrix}
4 \\
13 \\
8 \\
1
\end{bmatrix}

\end{align*}
$$

Basic properties of $s$ that I can prove:

* $ s \text{ is not linear} $
* $ s(\boldsymbol{0}) = \boldsymbol{0} $
* $ s(\lambda\boldsymbol{\theta}) = |\lambda|s(\boldsymbol{\theta}) $
* Let $\boldsymbol\theta' = \boldsymbol\theta - \theta_{min}$, where $\theta_{min}$ is the smallest component of $\boldsymbol\theta$. Then $s(\boldsymbol\theta)=s(\boldsymbol\theta') $

In my experiments, I observed that the repeated application of $s$ converged to the zero vector for all sample inputs I tried: millions of unique $\boldsymbol{\theta}\in {\mathbb{N}_0}^4$

As a purely recreational exercise, my goal is to prove or disprove that this holds *for all* $\boldsymbol{\theta}\in {\mathbb{N}_0}^4$:

$ \forall_{\boldsymbol{\theta}} \exists_{n\ge 0} \space \space \overbrace{s \circ \ s \circ \dots s}^{n} (\boldsymbol{\theta}) = \boldsymbol{0} $

Now my questions:

1. Is $s$ a common function in some domain or application?
1. Which research keyworks would you recommend to find more about $s$  (or similar function)?

Search expressions like "adjacent element distance" and variations thereof didn't help much. These often lead to pages more on the implementation/coding side of the problem.

This is purely a recreational problem. There is no real application that I know of - but I would like to know. I'm not a professional mathematician. My background is computer science.

Thank you