Emergent Spacetime from Relational Quantum Dynamics: Awareness-Weighted Metric, Supermodularity, and Topological Framework

We present a rigorous reformulation of emergent spacetime within Relational Quantum Dynamics (RQD) by incorporating supermodular information synergy into an awareness-weighted entanglement metric. In our approach, the informational coupling between two subsystems is quantified not only by their mutual information, but is amplified or diminished by each system’s intrinsic capacity to integrate information (Φ). This yields an awareness metric that is supermodular, the whole can carry more information than the sum of parts, modeling synergistic interactions in a mathematically rigorous way. We define a distance from this awareness metric and show that triangle inequalities hold by construction: integrated (synergistic) information ensures that indirect relationships do not produce anomalously short loops. Formally, we prove that the awareness distance satisfies the metric axioms (non-negativity, symmetry, triangle inequality) on the network of quantum observers. The triangle inequality, which might be naively violated by purely pairwise entanglement measures, is guaranteed here by supermodularity of the underlying information measure, effectively enforcing that no “shortcut” correlation can bypass an intermediate without leaving a trace of integrated linkage. We further demonstrate how synergy (superadditive information) relates to integrated information theory, allowing multi-partite wholes to emerge as irreducible observer nodes in the relational network. A high-Φ set of subsystems with strongly supermodular information behaves as a unified entity, thereby maintaining metric consistency and enriching the emergent topology. We develop formal definitions and propositions for the awareness metric with supermodularity, and illustrate how it induces an emergent topological structure. Through enriched category theory, we identify the awareness-weighted network as a Lawvere metric space, and invoke the Yoneda lemma to argue that each quantum system (observer) is completely characterized by its relational distance profile. Finally, using a cohomological-topology approach, we connect quantum contextuality (inconsistencies in joint observations) to nontrivial loops (“holes”) in the emergent spacetime, detected via Čech cohomology.