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.
changzheng zhou, ziqing zhou· Zenodo (CERN European Organi...· 0 citations
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.
changzheng zhou, ziqing zhou· Zenodo (CERN European Organi...· 0 citations
Cognitive systems face the interface problem between continuous perceptualinputs and discrete symbolic representations. This paper models the perceptionsymbol interface as a fiber bundle structure, with the perceptual manifold as thebase space and the candidate symbol distribution simplex as the fiber, and specifiesthe transport rules of symbol distributions along perceptual paths via a connection. The core findings include: the connection curvature exactly corresponds tolocal sequential effects in simply connected neighborhoods; the full holonomy groupdecomposes into a curvature-generated restricted holonomy component and a monodromy component given by the fundamental group of the base space, the formerbeing continuously tunable and the latter topologically stable. The discontinuity ofthe symbolization projection on the distribution simplex constitutes a boundary defect independent of curvature. Accordingly, we propose a geometric regularizationobjective comprising four terms—curvature penalty, small-loop constraint, largeloop symbol-level penalty, and task loss—providing design principles for neurosymbolic systems based on the dual mechanism of curvature and monodromy control. This framework unifies perceptual ambiguity, sequential effects, memory pathdependence, and framework incommensurability in the language of differential geometry, and establishes structural correspondence with existing neuro-symbolicmethods via sheaf cohomology
changzheng zhou, ziqing zhou· Zenodo (CERN European Organi...· 0 citations
Cognitive systems face the interface problem between continuous perceptualinputs and discrete symbolic representations. This paper models the perceptionsymbol interface as a fiber bundle structure, with the perceptual manifold as thebase space and the candidate symbol distribution simplex as the fiber, and specifiesthe transport rules of symbol distributions along perceptual paths via a connection. The core findings include: the connection curvature exactly corresponds tolocal sequential effects in simply connected neighborhoods; the full holonomy groupdecomposes into a curvature-generated restricted holonomy component and a monodromy component given by the fundamental group of the base space, the formerbeing continuously tunable and the latter topologically stable. The discontinuity ofthe symbolization projection on the distribution simplex constitutes a boundary defect independent of curvature. Accordingly, we propose a geometric regularizationobjective comprising four terms—curvature penalty, small-loop constraint, largeloop symbol-level penalty, and task loss—providing design principles for neurosymbolic systems based on the dual mechanism of curvature and monodromy control. This framework unifies perceptual ambiguity, sequential effects, memory pathdependence, and framework incommensurability in the language of differential geometry, and establishes structural correspondence with existing neuro-symbolicmethods via sheaf cohomology
changzheng zhou, ziqing zhou· Zenodo (CERN European Organi...· 0 citations
The evolution of scientific theories exhibits a pronounced hierarchical character,with each level providing effective predictions within specific operational boundaries, and undergoing structural transitions at these boundaries. Based on empirical patterns from the history of science and hard constraints from formal systems,this paper establishes a “structural network” model that abstracts a theory as anetwork structure with assumption nodes, logical dependency edges, and evidenceweights, and provides a systematic construction protocol for mapping theoreticaltexts to formal networks. The study shows that transitions between theoreticallevels are not smooth accumulations but are driven by four types of topologicalreconstruction mechanisms: coarse-grained embedding, hypothesis-space bifurcation, cross-domain bridging, and theory freezing. Level transitions are subject toa triple constraint of observational resolution, quantum uncertainty, and computational complexity, and empirically satisfy a structural information conservationcondition, namely that the network complexity of the new theory is not less thanthat of the old theory minus the reconstruction cost. This paper further proposesa three-dimensional assessment vector for operational completeness, transformingtheory choice from single‑metric optimisation into a cognitive‑economic decisionon the Pareto frontier, and uses spectral features to accelerate the screening. Theframework provides a computable formal description tool for phenomena such asscientific revolutions, theory deadlocks, and cross‑domain unification, and givestestable predictions through the Standard Model, the string landscape, and historical transition cases.
changzheng zhou, ziqing zhou· Zenodo (CERN European Organi...· 0 citations
The evolution of scientific theories exhibits a pronounced hierarchical character,with each level providing effective predictions within specific operational boundaries, and undergoing structural transitions at these boundaries. Based on empirical patterns from the history of science and hard constraints from formal systems,this paper establishes a “structural network” model that abstracts a theory as anetwork structure with assumption nodes, logical dependency edges, and evidenceweights, and provides a systematic construction protocol for mapping theoreticaltexts to formal networks. The study shows that transitions between theoreticallevels are not smooth accumulations but are driven by four types of topologicalreconstruction mechanisms: coarse-grained embedding, hypothesis-space bifurcation, cross-domain bridging, and theory freezing. Level transitions are subject toa triple constraint of observational resolution, quantum uncertainty, and computational complexity, and empirically satisfy a structural information conservationcondition, namely that the network complexity of the new theory is not less thanthat of the old theory minus the reconstruction cost. This paper further proposesa three-dimensional assessment vector for operational completeness, transformingtheory choice from single‑metric optimisation into a cognitive‑economic decisionon the Pareto frontier, and uses spectral features to accelerate the screening. Theframework provides a computable formal description tool for phenomena such asscientific revolutions, theory deadlocks, and cross‑domain unification, and givestestable predictions through the Standard Model, the string landscape, and historical transition cases.
changzheng zhou, ziqing zhou· Zenodo (CERN European Organi...· 0 citations
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