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，相关参与者的能力都无法预先获知。无论接收者是谁，消息的时空尺度可能过快、过慢、过大或过小，导致永远无法解码。因此，设计一种与时空尺度无关的消息编码方式具有重要意义。本文利用分形函数的自相似性和标度不变性，提出将其作为自执行无限频率载波用于消息传递，称为"分形消息传递"。从空间嵌入出发，我们引入了一种面向时空无标度消息传递的框架。在考虑与时空无关的消息传输框架时，有趣的是能够编码消息使其在多个时空尺度上被解码。因此，本文提出的核心框架是：将二进制消息编码为具有幂率分布特征的无限频率波与幅值，传输该消息后进行解码与重构。通过使用已知分形函数——魏尔斯特拉斯函数的各个分量作为消息载波，对每个分量的幅值进行调制以嵌入二进制数据流，从而实现与时空无关的消息传递方法。