Complex systems can be described at myriad different scales, and their causal workings often have multiscale structure (e.g., a computer can be described at the microscale of its hardware circuitry, the mesoscale of its machine code, and the macroscale of its operating system). While scientists study and model systems across the full hierarchy of their scales, from microphysics to macroeconomics, there is debate about what the macroscales of systems can possibly add beyond mere compression. To resolve this longstanding issue, here a new theory of emergence is introduced wherein the different scales of a system are treated like slices of a higher-dimensional object. The theory can distinguish which of these scales possess unique causal contributions, and which are not causally relevant. Constructed from an axiomatic notion of causation, the theory's application is demonstrated in coarse-grains of Markov chains. It identifies all cases of macroscale causation: instances where reduction to a microscale is possible, yet lossy about causation. Furthermore, the theory posits a causal apportioning schema that calculates the causal contribution of each scale, showing what each uniquely adds. Finally, it reveals a novel measure of emergent complexity: how widely distributed a system's causal workings are across its hierarchy of scales.

翻译：复杂系统可以在无数不同的尺度上进行描述，其因果运作通常具有多尺度结构（例如，计算机可以在其硬件电路的微观尺度、机器代码的中观尺度以及操作系统的宏观尺度上进行描述）。尽管科学家们从微观物理学到宏观经济学，跨越系统尺度的完整层次结构来研究和建模系统，但对于系统的宏观尺度除了单纯的压缩之外还能提供什么，一直存在争论。为了解决这一长期存在的问题，本文引入了一种新的涌现理论，其中系统的不同尺度被视为一个高维物体的切片。该理论能够区分哪些尺度具有独特的因果贡献，哪些在因果上不相关。该理论基于因果关系的公理化概念构建，并在马尔可夫链的粗粒化中演示了其应用。它识别了所有宏观尺度因果作用的案例：即那些可以还原到微观尺度，但在因果关系上存在信息损失的情况。此外，该理论提出了一种因果分配方案，用于计算每个尺度的因果贡献，展示每个尺度独特添加了什么。最后，它揭示了一种新颖的涌现复杂性度量：即系统的因果运作在其尺度层次结构中分布的范围有多广。