arXiv:2507.03065v2 Announce Type: replace Abstract: Why do some macroscopic structures remain identifiable even though their microscopic constituents continually change? Vortices persist while fluid parcels turn over, neural memories persist while spikes and synapses fluctuate, and institutions persist while individuals enter and leave. We propose a scale-relative answer: an emergent property is a persistent nontrivial homology class $[z]\in H_p=\ker\partial_p/\im\partial_{p+1}$, a macro-feature that is closed but not exact across a filtration of descriptions. This identification turns emergen
The paper was just published, representing a theoretical advancement in understanding emergent properties, particularly relevant to complex systems and AI research.
This theoretical framework offers a novel mathematical lens to understand and predict 'persistence' in complex systems, which can significantly influence the design and analysis of advanced AI, biological, and social models.
The proposed theory provides a new formal language for identifying and quantifying emergent structures, potentially shifting how researchers approach system design and analysis in fields from AI to sociology.
- · AI researchers
- · Complex systems theorists
- · Biological modeling
- · Topology and applied mathematics
- · Reductionist approaches to complex systems
The paper provides a new mathematical tool, persistent homology, for characterizing how macroscopic structures emerge and persist despite microscopic changes.
This framework could lead to breakthroughs in designing more robust and adaptive AI systems that can identify and maintain functions amidst dynamic environments.
A deeper understanding of emergent properties might enable the creation of truly autonomous and self-organizing AI agents with complex, persistent behaviors.
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Read at arXiv cs.LG