Persistent Homology Framework Proposed as Mathematical Theory of Emergent Structure
Researchers propose that emergent macroscopic structures can be mathematically defined as persistent nontrivial homology classes that remain stable despite changing microscopic constituents. The framework uses topological methods to identify when macro-features are closed but not exact across different scales of description. This approach unifies six signatures of emergence and generates testable predictions across atmospheric, neural, and social systems.
A new theoretical framework uses persistent homology—a mathematical tool from algebraic topology—to explain why macroscopic structures like vortices, neural memories, and institutions persist despite continuous change in their underlying components. The authors define emergent properties as persistent homology classes that are closed but not exact within a filtration of descriptions, converting emergence into a measurement problem. The scaffold-flow framework employs Hodge decomposition to separate harmonic macro-level features from micro-level flows, and introduces a contractive-similarity graph operator to predict structural robustness. The theory unifies six established signatures of emergence (inevitability, coherence, irreducibility, complementarity, robustness, and hierarchy) within a single mathematical language. Importantly, the framework generates falsifiable predictions: genuine emergent structures should persist across filtrations, remain spectrally stable, respond disproportionately to harmonic interventions, and require timescale separation for hierarchical autonomy.
What's missing
The paper does not discuss computational complexity or practical algorithmic implementation for applying persistent homology to real-world systems at scale. Additionally, while the framework makes falsifiable predictions, the paper does not present empirical validation results from atmospheric, neural, or social systems—these appear to be proposed future tests rather than completed demonstrations.
What different sources said
- arXiv cs.LGCenter
Persistent Homology as a Theory of Emergent Structure
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