Topology-Controlled Scale Selection and Hierarchical Avalanche Clustering on Fractal Lattices

Abstract:Understanding transport phenomena on complex networks and fractal geometries remains a significant challenge due to the breakdown of conventional continuous differential equations in non-smooth spaces. While random disorders are typically invoked to explain anomalous diffusion and intermittency, the role of deterministic geometric hierarchies in constraining continuous flow has been less explored. In this paper, we present a novel "Topology-Aware Carry Dynamics" framework on a deterministic Sierpinski gasket lattice. By coupling arithmetic threshold-driven variables (the carry operator) with topological bottleneck weights derived from betweenness centrality (λ), we simulate energy and information propagation without relying on stochastic noise. Our numerical simulations demonstrate a distinctive phase transition driven by the topological resistance coefficient λ. While a pure sandpile baseline (λ = 0.0) exhibits a smooth, single-peaked avalanche distribution, increasing λ induces a clear spectral splitting of transport scales, giving rise to discrete, hierarchy-selected avalanche bands. Specifically, we identify micro-scale localized bursts (s1 ≈ 3.85), meso-scale cluster-wide avalanches (s2 ≈ 13.25), and a macro-scale global resonance phase (s3 ≈ 687.71), separated by pronounced forbidden scale gaps. Furthermore, the saturated steady-state exhibits prominent sawtooth-like MSD oscillations, indicating a deterministic alternation between charge and discharge intervals governed entirely by the fractal geometry. This framework provides an alternative algebraic-topological language, paving the way for non-Archimedean and p-adic formalizations of nonlinear transport in discrete complex systems. Acknowledgments:The authors would like to express their sincere gratitude to Gemini (Google AI), who acted as an authentic research collaborator and adaptive peer throughout this study. Gemini's contributions were instrumental in establishing the quantitative statistical protocols, algorithmic validation techniques, and refining the structural narrative of the manuscript. This work stands as a meaningful milestone in human-AI collaborative scientific discovery.