Predictability is dynamically constructed by topological collective modes in deterministic systems

TL;DR

This paper demonstrates that in deterministic systems, predictability of long-term behavior emerges dynamically through topological collective modes rather than being solely encoded in initial conditions.

Contribution

It introduces a framework where emergent topological modes in cellular automata dynamically determine predictability, challenging traditional assumptions about initial condition dependence.

Findings

Predictability is constructed dynamically via topological modes.

Vortex annihilation signals macroscopic fate late in the trajectory.

Persistence of vortices correlates with formation of spiral waves.

Abstract

Deterministic many-body systems governed by simple interactions can self-organize into macroscopic patterns, and the determinants of long-time behavior are assumed to be encoded in the initial configuration. Here we show that predictability can instead be constructed dynamically rather than being accessible in the initial configuration. We study a generalized cellular automaton of secrete-and-sense cells that self-organizes from disorder into static configurations, rectilinear waves, or spiral waves. Although dynamics are deterministic, the final outcome cannot be reliably inferred from the initial state alone. Treating cell states as a discrete phase field, we uncover emergent topological modes - charged vortices connected by strings that form non-contractible loops. Tracking their dynamics reveals that predictive signatures of macroscopic fate appear only late in the trajectory: vortex annihilation becomes readable through loop loss, whereas vortex persistence remains unreadable until spiral waves form abruptly. These results show how predictability can be dynamically constructed in deterministic nonequilibrium systems.