Predictability can be dynamically constructed in deterministic systems

Abstract

A deterministic system's future can be fixed from the start yet remain unpredictable because of chaos or computational irreducibility. Here we reveal another source of unpredictability for non-chaotic deterministic systems: collective entities making the future legible are initially absent but dynamically constructed. In a generalized cellular automaton, disordered lattices evolved into static configurations, rectilinear waves, or spiral waves. Although initial configurations fixed each fate, machine-learning models did no better than randomly guessing fates from them. Recoding cell states geometrically revealed self-organization of prediction-enabling topological entities: vortices, non-contractible-loop strings, and, most generally, a winding field describing how same-state regions wrap the lattice. As the winding field self-organized, initially unpredictable fates became increasingly legible. Static and rectilinear-wave fates became progressively predictable, whereas spiral-wave fate became accurately predictable only near wave formation. These results establish that self-organization can build not merely order but entities that make certain futures visible, revealing a gap between determinism and practical predictability.