Geometric origin of hierarchical patterns through history-dependent growth and arrest

Hierarchical patterns are widespread in natural and synthetic systems, yet their formation is most often interpreted through diffusion-driven instabilities or static geometric constructions. While successful in many contexts, these frameworks struggle to encode growth history, temporal asymmetry, and pattern formation in non-diffusive, out-of-equilibrium systems. Here, I introduce the TRACE model (Temporal Recording through Arrested Coupled Evolution), a minimal framework in which pattern formation emerges from the competition between front propagation and kinetic arrest, without invoking diffusive transport. Growing domains leave geometric footprints that record their individual growth histories through exclusion. Numerical simulations reveal a broad spectrum of emergent patterns—including cell-like domains, stripes, banded structures, and hierarchical boundaries—arising from variations in spatial confinement, nucleation timing, and domain density. Crucially, the number and scale of patterns depend on accumulated domain interactions rather than nucleation alone. Finally, the model qualitatively reproduces key features observed in a minimal two-component chemical system exhibiting hierarchical macroscopic pattern formation, establishing a diffusion-free route to hierarchy encoded as a geometric record of growth history.