  Single-file transport, where particles diffuse in narrow channels while not
overtaking each other, is a fundamental model for the tracer subdiffusion
observed in confined systems, such as zeolites or carbon nanotubes. This
anomalous behavior originates from strong bath-tracer correlations in 1D,
which, despite extensive effort, have however remained elusive, because they
involve an infinite hierarchy of equations. Here, for the Symmetric Exclusion
Process, a paradigmatic model of single-file diffusion, we break the hierarchy
and unveil a closed exact equation satisfied by these correlations, which we
solve. Beyond quantifying the correlations, the central role of this key
equation as a novel tool for interacting particle systems is further
demonstrated by showing that it applies to out-of equilibrium situations, other
observables and other representative single-file systems.
