Persistent Homology Framework Proposed as Mathematical Theory of Emergent Structure
A preprint posted to arXiv proposes using persistent homology — a tool from algebraic topology — to formally define and measure emergent structures in complex systems. The framework aims to explain why large-scale patterns like vortices, neural memories, and social institutions remain stable even as their underlying components constantly change. If validated, the approach could provide a unified mathematical language for studying emergence across physics, neuroscience, and social science.
Researchers have submitted a preprint to arXiv arguing that emergent properties in complex systems can be rigorously identified as persistent nontrivial homology classes within a mathematical filtration of descriptions. The core idea is that a macro-feature qualifies as genuinely emergent if it is 'closed but not exact' — meaning it cannot be reduced to or derived from the micro-level dynamics beneath it. The paper introduces a contractive-similarity (CS) graph operator to quantify the robustness of these structures via spectral gaps, and applies Hodge decomposition to separate macro-level 'scaffold' features from micro-level 'flow.' The framework is claimed to capture six canonical signatures of emergence — inevitability, coherence, irreducibility, complementarity, robustness, and hierarchy — within a single formalism. The authors offer falsifiable predictions: genuine emergent structures should persist across filtrations, remain spectrally stable, respond disproportionately to harmonic interventions, and require timescale separation for hierarchical autonomy. The work spans applications in atmospheric science, neuroscience, and social systems, though it remains a preprint and has not yet undergone formal peer review.
What's missing
As a preprint, this work has not undergone peer review, and no empirical validation of the framework's predictions is presented within the paper itself. It is also unclear whether the proposed falsifiable predictions have been tested against existing datasets, or whether the framework has been compared against alternative mathematical theories of emergence.
What different sources said
- arXiv cs.LGCenter
Persistent Homology as a Theory of Emergent Structure
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