Coexistence Compatibility as a Constraint on Realizability: A Spatio-Temporal Framework for Complex Systems

This preprint introduces a spatio-temporal extension of compatibility in complex systems. Traditional formulations treat compatibility as a static structural property. However, many configurations that are structurally admissible fail to be realized in practice due to mismatches in characteristic timescales between interacting components. We formalize this limitation by introducing temporal incompatibility and defining a dynamic Congruity Index (ICSD), extending the structural formulation (ICSS). We further introduce the realization gap (GR) as a quantitative measure of the discrepancy between theoretical admissibility and effective realizability. A minimal toy model demonstrates that increasing temporal mismatch leads to a systematic reduction in dynamic admissibility and the emergence of a realization gap, even when structural compatibility remains unchanged. Importantly, the framework reveals the existence of a critical threshold beyond which configurations transition from dynamically realizable to dynamically inadmissible. This shows that incompatibility is not only gradual but can exhibit threshold-like behavior. These results suggest that coexistence compatibility is inherently spatio-temporal and that realizability requires both structural consistency and temporal alignment. This work provides a minimal and general framework for analyzing dynamic admissibility in complex systems, with potential applications in multi-agent systems, distributed computing, physical systems, and adaptive decision-making architectures.