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| Format: | Recurso digital |
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Zenodo
2025
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| Online Access: | https://doi.org/10.5281/zenodo.17873455 |
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Table of Contents:
- <p><span>Complex adaptive systems (CAS) are organised into multiple emergent orders in which</span></p> <p><span>qualitatively distinct patterns, constraints, and stability regimes appear at successive scales.</span></p> <p><span>Although such multi-level structure is ubiquitous across biological, ecological, and socio-technical</span></p> <p><span>domains, there is no general mathematical account of why higher organisational orders differ in</span></p> <p><span>their effective dimensionality, or how this dimensionality shapes their dynamical behaviour.</span></p> <p><span>We model an emergent order as the image of a coarse-graining map</span></p> <p><span>π</span><span>k </span><span>: R</span><span>N</span><span>k−1 </span><span>→R</span><span>N</span><span>k</span></p> <p><span>,</span></p> <p><span>which compresses a high-dimensional microstate into a lower-dimensional macrostate. Iterating</span></p> <p><span>these maps generates a hierarchy</span></p> <p><span>S</span><span>0 </span><span>→S</span><span>1 </span><span>→···→S</span><span>K </span><span>,</span></p> <p><span>where each order-</span><span>k </span><span>system has an effective dimensionality </span><span>N</span><span>k </span><span>and evolves on a manifold </span><span>M</span><span>k </span><span>⊆</span></p> <p><span>R</span><span>N</span><span>k </span><span>endowed with its own stability geometry. We show that when the coarse-graining maps are</span></p> <p><span>non-invertible—the generic case for aggregation and information loss—the effective dimensionality</span></p> <p><span>decreases strictly across organisational orders, N</span><span>0 </span><span>>N</span><span>1 </span><span>>···>N</span><span>K</span><span>.</span></p> <p><span>This dimensionality collapse provides a structural explanation for why emergent orders display</span></p> <p><span>distinct curvature, coupling structure, and canonical instability modes, and why micro-level</span></p> <p><span>laws do not uniquely determine higher-order stability properties. The framework is intentionally</span></p> <p><span>minimal and substrate-agnostic, offering a general account of how multi-level organisation, the</span></p> <p><span>limits of reductionism, and scale-dependent resilience arise from the systematic compression of</span></p> <p><span>micro-dynamics into coarse-grained macro-variables.</span></p>