Scale-Dependent Emergence of Spatial Connectivity from Trace-Based Dynamics

We investigated the emergence of spatial connectivity from a minimal pre-geometric model based on interaction, extinction, and trace reinforcement dynamics in order to examine whether stable spatial regularities can emerge even in the absence of predefined mathematical laws, geometric structure, or known physical particles and gauge fields. Here we present a reproducible minimal model of relational organization exhibiting scale-dependent emergent dynamics. In our model, elements leave relational traces when they disappear. Repeated traces reinforce effective proximity between surviving elements, allowing large-scale connectivity to emerge without assuming predefined geometry, spatial coordinates, or particle-based interactions. The resulting proximity network exhibits a sharp connectivity transition as a function of the trace reinforcement threshold k. For sufficiently small and intermediate system sizes, all simulations reproducibly selected a critical threshold kc = 3 across broad parameter ranges and random initial conditions. However, larger systems exhibited a crossover toward a distinct regime characterized by kc = 2, with an intermediate coexistence region in which both kc = 2 and kc = 3 realizations were observed. To further characterize the transition, we estimated the effective information compression associated with the emergence of the proximity network. While the absolute information reduction increased approximately with system size, the relative compression ratio exhibited a pronounced dip near the crossover region between the competing reinforcement regimes. These results indicate that the observed transition is not a consequence of fine parameter tuning, but instead reflects scale-dependent emergent dynamics intrinsic to the history-reinforcement dynamics itself. The coexistence of competing critical regimes together with distinct information-compression structures suggests that emergent spatial organization may possess multiple reinforcement phases depending on system scale and relational stability conditions. At high reinforcement thresholds, a small number of strongly reinforced long-range relational structures persisted even after the collapse of large-scale connectivity, suggesting that history-dependent correlations may remain dynamically stable beyond the stability limit of the emergent spatial phase. More broadly, the present framework supports the possibility that effective spatial structure, large-scale information organization, and potentially aspects of physical regularity may emerge dynamically from history-dependent relational reinforcement rather than from predefined geometric or mathematical principles.