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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.12016 |
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| _version_ | 1866908491989385216 |
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| author | Chen, Liang |
| author_facet | Chen, Liang |
| contents | Complex systems universally exhibit emergence, where macroscopic dynamics arise from local interactions, but a predictive law governing this process has been absent. We establish and verify such a law. We define a system's causal power at a spatial scale, $\ell$, as its Effective Information (EI$_\ell$), measured by the mutual information between a targeted, maximum-entropy intervention and its outcome. From this, we derive and prove a Middle-Scale Peak Theorem: for a broad class of systems with local interactions, EI$_\ell$ is not monotonic but exhibits a strict maximum at a mesoscopic scale $\ell^*$. This peak is a necessary consequence of a fundamental trade-off between noise-averaging at small scales and locality-limited response at large scales. We provide quantitative, reproducible evidence for this law in two distinct domains: a 2D Ising model near criticality and a model of agent-based collective behavior. In both systems, the predicted unimodal peak is decisively confirmed by statistical model selection. Our work establishes a falsifiable, first-principles law that identifies the natural scale of emergence, providing a quantitative foundation for the discovery of effective theories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_12016 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Law of Emergence: Maximum Causal Power at the Mesoscale Chen, Liang Information Theory Complex systems universally exhibit emergence, where macroscopic dynamics arise from local interactions, but a predictive law governing this process has been absent. We establish and verify such a law. We define a system's causal power at a spatial scale, $\ell$, as its Effective Information (EI$_\ell$), measured by the mutual information between a targeted, maximum-entropy intervention and its outcome. From this, we derive and prove a Middle-Scale Peak Theorem: for a broad class of systems with local interactions, EI$_\ell$ is not monotonic but exhibits a strict maximum at a mesoscopic scale $\ell^*$. This peak is a necessary consequence of a fundamental trade-off between noise-averaging at small scales and locality-limited response at large scales. We provide quantitative, reproducible evidence for this law in two distinct domains: a 2D Ising model near criticality and a model of agent-based collective behavior. In both systems, the predicted unimodal peak is decisively confirmed by statistical model selection. Our work establishes a falsifiable, first-principles law that identifies the natural scale of emergence, providing a quantitative foundation for the discovery of effective theories. |
| title | A Law of Emergence: Maximum Causal Power at the Mesoscale |
| topic | Information Theory |
| url | https://arxiv.org/abs/2508.12016 |