Stable memories maintained by local rules require non-local correlations

Stable cellular memories, such as heritable histone‐modification patterns, in noisy environments raise a general question: how can local Hebbian-like rules maintain long-lived information in noisy biological systems? Recent work shows that coupling between 3D genome architecture and histone reader–writer dynamics, together with limits on reader–writer activity, can stabilize epigenetic states. Motivated by this, we develop a general framework for stable memory in physical and biological systems where interactions are reinforced by a local, Hebbian-like rule driven by correlations in the physical degrees of freedom generated by the system's own dynamics. When these correlations are sufficiently non-local, missing or perturbed interactions can be reconstructed from correlations among distant sites, yielding robust memory. Vulnerabilities arise from coordinated deletions that simultaneously remove the correlation pathways supporting a given interaction; which joint deletions are dangerous is dictated by the mapping from interactions to correlations. We instantiate this framework in three settings, the Hopfield model, tile self-assembly, and a polymer model of chromatin, demonstrating how non-local correlations induced by physics enable locally maintained, stable memories and delineating their failure modes.