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 $\boldsymbol{\theta}$:

$ \forall_{\boldsymbol{\theta} \in {\mathbb{N}_0}^4} \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?
2. 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