Self-Organizing Graph Reasoning Evolves into a Critical State for Continuous Discovery Through Structural-Semantic Dynamics

We report fundamental insights into how agentic graph reasoning systems spontaneously evolve toward a critical state that sustains continuous semantic discovery. By rigorously analyzing structural (Von Neumann graph entropy) and semantic (embedding) entropy, we identify a subtle yet robust regime in which semantic entropy persistently dominates over structural entropy. This interplay is quantified by a dimensionless Critical Discovery Parameter that stabilizes at a small negative value, indicating a consistent excess of semantic entropy. Empirically, we observe a stable fraction (12%) of "surprising" edges, links between semantically distant concepts, providing evidence of long-range or cross-domain connections that drive continuous innovation. Concomitantly, the system exhibits scale-free and small-world topological features, alongside a negative cross-correlation between structural and semantic measures, reinforcing the analogy to self-organized criticality. These results establish clear parallels with critical phenomena in physical, biological, and cognitive complex systems, revealing an entropy-based principle governing adaptability and continuous innovation. Crucially, semantic richness emerges as the underlying driver of sustained exploration, despite not being explicitly used by the reasoning process. Our findings provide interdisciplinary insights and practical strategies for engineering intelligent systems with intrinsic capacities for long-term discovery and adaptation, and offer insights into how model training strategies can be developed that reinforce critical discovery.