Cognitive Shell B: Persistent Homology of Cognitive Phase Transitions——Birth-Death Analysis of Topological Features on Filtered Complexes

Abstract

The sudden structural transformations experienced by cognitive systems duringevolution are difficult to characterise with traditional continuous dynamics frameworks. This paper constructs a mathematical framework that maps cognitive statespaces to sequences of filtered simplicial complexes, and employs persistent homology to track the birth and death of connected components, one-dimensional cycles,and higher-dimensional cavities as the filtration parameter varies. The core workoperates on two levels. At the descriptive level, we establish a rigorous constructionfrom concept-association weights to clique-complex filtrations, and encode cognitive structures at multiple scales via barcodes. At the causal attribution level, weintroduce order-parameter dynamics, distinguish scanning events produced by theanalyst from crossing events produced by the system’s own evolution, and provethat under scale separation and genericity conditions, every effective crossing eventcorresponds to a birth-death endpoint in the barcode of a weight-matrix snapshot(surjective correspondence), and that the correspondence is bijective on the subclass of trajectories where the order parameter evolves effectively monotonically;for noise-driven recurrent crossings, we give a minimal resolution-time convention to make crossing events well-defined, and establish rewiring multiplicity asan observable characterising metastability near the critical neighbourhood. Onthis basis, we provide statistical significance criteria for candidate phase-transitionevents, including bridge-edge and modularity-jump tests for zero-dimensional integration, representative-cycle and persistence criteria for one-dimensional feedbackloops, and a structural-tension criterion for higher-dimensional cavities——the lastaccompanied by a positive-persistence demonstration example as an operational illustration. We further propose a falsifiable conjecture: creative insight correspondstopologically to a crossing event of a long-lived one-dimensional homology classthat passes the significance tests, and the coincidence condition for the insight moment has been fully operationalised as executable statistical tests. This frameworkprovides computable tools for quantifying cognitive creativity, predicting criticaltransitions in belief systems, and designing intelligent systems with topologicalself-awareness.