Cognitive Shell E: Hodge Decomposition of Causal Emergence– Spectral Triple Decomposition of Information Flow on Constraint Networks
Abstract
The quantification of causal emergence faces a fundamental difficulty: how canmacroscopic causal power be identified in microscopic dynamics without falling intoa circular reliance on a presupposed coarse-graining strategy? In this paper, theconstraint network of a cognitive system is modelled as a simplicial complex, anddiscrete Hodge decomposition is introduced as an analytical tool to rigorously decompose the edge information flow into three orthogonal components: the gradientfield (corresponding to driving causal effects), the curl field (corresponding to cyclicfeedback structures), and the harmonic field (corresponding to global emergent patterns). Under the spectral definition (Definition 4.1), causal emergence manifestsas a sudden change in the harmonic energy proportion at a critical threshold of thespectral parameter (Corollary 4.1), while the strict equivalence between spectralevents and topological events is given by Theorem 4.1. This framework unifiesthe theory of effective information and the theory of integrated information on thesame spectral-analytic base, and provides a computable algorithmic pathway fordetecting critical points of emergence from observational data. On this basis, we introduce slow-scale operator dynamics, which closes the three links of directionality,estimation consistency, and dynamical origin of the spectral criterion, making thedetermination of causal emergence an operational spectral-event detection protocol.
