In many communication contexts, the capabilities of the involved actors cannot be known beforehand, whether it is a cell, a plant, an insect, or even a life form unknown to Earth. Regardless of the recipient, the message space and time scale could be too fast, too slow, too large, or too small and may never be decoded. Therefore, it pays to devise a way to encode messages agnostic of space and time scales. We propose the use of fractal functions as self-executable infinite-frequency carriers for sending messages, given their properties of structural self-similarity and scale invariance. We call it `fractal messaging'. Starting from a spatial embedding, we introduce a framework for a space-time scale-free messaging approach to this challenge. When considering a space and time-agnostic framework for message transmission, it would be interesting to encode a message such that it could be decoded at several spatio-temporal scales. Hence, the core idea of the framework proposed herein is to encode a binary message as waves along infinitely many frequencies (in power-like distributions) and amplitudes, transmit such a message, and then decode and reproduce it. To do so, the components of the Weierstrass function, a known fractal, are used as carriers of the message. Each component will have its amplitude modulated to embed the binary stream, allowing for a space-time-agnostic approach to messaging.

翻译：在许多通信场景中，无论是细胞、植物、昆虫，甚至地球未知的生命形式，通信参与者的能力都无法预先知晓。无论接收方是谁，消息的时空尺度可能过快、过慢、过大或过小，从而永远无法被解码。因此，设计一种与时空尺度无关的消息编码方法具有重要意义。鉴于分形函数具有结构自相似性和尺度不变性的特性，我们提出将其用作自执行无限频率载波来传输消息，并将其命名为“分形消息传递”。从空间嵌入出发，我们引入了一种用于解决这一挑战的时空无标度消息传递框架。在考虑与时空无关的消息传输框架时，若能对消息进行编码使其可在多个时空尺度上被解码，将极具价值。因此，本文提出的框架核心思想是：将二进制消息编码为沿无限频率（呈幂率分布）和振幅分布的波，传输该消息，随后对其进行解码并重构。为实现这一目标，我们利用已知分形——魏尔斯特拉斯函数的各个分量作为消息载波，对每个分量的振幅进行调制以嵌入二进制比特流，从而构建一种与时空无关的消息传递方法。