From Synchronization to Coherence: A Relational Topology for Emergent Systems — Revised 2PS Topology and λᵣel Validation

This work presents a novel interpretation of a nine-node relational structure as a dynamic topology of relational coherence in complex systems. Moving beyond traditional symbolic or psychological interpretations, the structure is formalized as a minimal architecture capable of supporting distributed synchronization, multilayer feedback, structural redundancy, and emergent global order. Building upon the Kuramoto model and its extensions to complex networks, the study argues that classical star-topology synchronization represents only a restricted case within a broader relational framework. The proposed 2PS topology extends this paradigm by integrating radial, circular, and cross-layer couplings, allowing coherence to emerge from structured interactions rather than from centralized control. The revised version adopts an official nine-node 2PS topology composed of one central coherence attractor, denoted by lambda (λ), and eight peripheral relational domains. The topology includes radial links from the central node, circular links among neighboring peripheral nodes, and a defined set of cross-layer connections: (1, 3), (2, 5), (2, 7), (2, 8), (4, 6), (4, 7), and (6, 8). This structure is proposed as a minimal relational blueprint for sustaining coherence under perturbation. The work introduces a relational coherence metric, λᵣel (t), combining phase alignment, structural connectivity, and information organization into a unified dynamic quantity. In the revised formulation, λᵣel (t) is expressed as a weighted integration of the Kuramoto order parameter R (t), a normalized algebraic connectivity term C (t), and an information-related term I (t). This enables a conceptual and operational transition from synchronization, understood as phase alignment, to coherence, understood as structured relational integration. The model is further explored in relation to natural and artificial systems, including cosmic filamentary structures, neural networks, wave-interference phenomena, toroidal or vortical flows, artificial intelligence architectures, and resilient infrastructure networks. In these systems, coherence is interpreted as a higher-order emergent property arising from relational geometry rather than from purely local interactions. Numerical simulations compare classical star-topology synchronization with the proposed 2PS relational topology under hub-failure conditions. The results suggest that while star-like architectures may achieve rapid initial synchronization, they are structurally fragile when centralized coordination fails. In contrast, the 2PS topology preserves substantially higher relational coherence through circular and cross-layer pathways. Additional validation using an IEEE-118-inspired network scenario further supports the interpretation of 2PS as a candidate design principle for resilient distributed systems. Rather than claiming that nature literally exhibits an eneagram-shaped geometry, this work interprets the nine-node topology as an abstract dynamical model describing how energy, information, and structure may propagate across complex systems. The framework establishes a conceptual bridge between synchronization theory, network dynamics, emergence-based models, and relational systems theory, offering a new perspective on coherence as a fundamental organizing principle across physical, biological, cognitive, artificial, and infrastructural domains.