Information-Theoretic Bounds and Relational Persistence in Chaotic Trajectory Recovery

This work investigates the fundamental information-theoretic limits of trajectory and identity recovery in chaotic Hamiltonian flows when initial tracking labels are absent. We examine how much memory is fundamentally required to recover particle histories after chaotic mixing has pushed trajectories beyond the predictability horizon. In particular, we investigate whether low-dimensional compressed sketches can reconstruct correspondences across scrambled ensembles, and whether collective multi-body relationships preserve ancestral origin signatures that individual laboratory coordinates appear to destroy. We establish that identity recovery under random projective sketching is governed by an exact Shannon channel-capacity bound rooted in geometric concentration of measure, analytically ruling out linear-memory conjectures. We demonstrate that recoverable identity in compact chaotic flows decays according to a universal power law in time, rather than an exponential drop, resolving longstanding parameter mismatches in phenomenological modeling. In coupled gravitational systems, we show that while scalar reductions fail due to virial crowding, the complete local relational network and topological neighborhood overlap retain ancestral identity long after Cartesian tracking collapses. These principles are corroborated across discrete symplectic maps, continuous self-gravitating clusters, and experimental solution-NMR protein dynamics under thermal fluctuations. The framework has potential implications for space situational awareness, sparse radar-tracklet association, computational biophysics and molecular-dynamics compression, galactic archaeology and the reconstruction of disrupted stellar streams, as well as information security and hardware analysis, including theoretical limits on structural leakage from continuous chaotic ciphers and multi-probe physical side channels. More broadly, the results suggest that chaotic dynamics do not necessarily erase identity information uniformly: while individual trajectories and Cartesian coordinates may lose ancestral correspondence, higher-order relational and topological structures can preserve recoverable information beyond conventional predictability horizons.