Figure depicts an idealised underdamped hierarchy structure with three Matsubara axes (x, y, z); to be generalised below. In reality these structures would be much more complex, generally with dimensions > 3, particularly for structured spectral densities. Within this diagram a) presents the movement of information away from the density matrix, ρ 0,0,0 , with each subplot representing two-dimensional slices across the xy-plane for a different value of z. Similarly b) shows information movement back towards the density matrix. Each face of the cubes which represent the auxiliary density operators are coloured based on whether they permit a transfer of virtual information in that particular direction. In addition, the cross-hatching of some ADOs denotes that they are governed by the alternative terminating equation of motion and will be unable to transfer non-Markovian virtual information beyond this boundary. In the discussion that follows we refer to ADOs increasing in tier as they move out along a Matsubara axis, away from the density matrix.