Awareness-Weighted Entanglement Metric and Emergent Topology in Relational Quantum Dynamics

This paper presents a novel theoretical framework wherein spatial geometry and topological structures emerge naturally from quantum information exchange within the Relational Quantum Dynamics (RQD) paradigm. Extending prior work on geometry extraction from quantum entanglement, we introduce an awareness-weighted entanglement metric, combining quantum mutual information and integrated information—a measure reflecting the internal coherence and holism of subsystems—to quantify relational proximity among quantum systems. This new metric, termed the "awareness metric," characterizes how strongly two subsystems are mutually aware based not only on their entanglement but also on their capacity for internal information integration. Using this metric, we construct an emergent relational distance that fulfills standard metric axioms, enabling the definition of a well-formed relational geometry. We demonstrate how this awareness-based distance naturally generates topological structures: disconnected sets of subsystems correspond to informationally isolated components, and contextual quantum correlations yield nontrivial loops and topological cycles in the emergent relational space. Integrated information identifies strongly integrated clusters within this structure, suggesting the existence of higher-level composite subsystems that function as unified wholes, thus enabling a multi-scale hierarchical relational description. We further formalize this relational geometry within an enriched categorical setting, showing that it constitutes a Lawvere metric space where distances between objects correspond directly to their mutual informational relationships. By employing the Yoneda lemma, we establish that each subsystem's identity emerges solely through its informational relations to others, aligning closely with RQD’s philosophical stance that systems possess no intrinsic, observer-independent existence. Additionally, we explore the relationship between quantum contextuality and topology, demonstrating how the impossibility of a globally consistent joint informational description manifests as topological obstructions—"holes"—in the relational geometry.