Identifiability of a linear dynamical reaction-diffusion system with directed interactions from the second-order statistics of a single snapshot

A single spatial snapshot of a system with multiple interacting populations captures positions but not the underlying dynamics that have led to the observation. This work asks what the stationary second-order spatial statistics of a snapshot can reveal about the structure of the linear reaction-diffusion dynamical model, and in particular about the pairwise directed entries of its interaction operator. The observation is the matrix-valued structure factor S(k) as a one-parameter family in the diffusion-inherited dependence on the wavenumber κ=k squared. The main result is that the directed interaction operator A and the diffusivity matrix D are generically identifiable for any number of interacting populations from S(κ) up to a single global time rescaling. The findings do not contradict known bounds and the recent result that spatial observational data only determine the Markov equivalence class of the underlying graph. In this work, we identify physical assumptions that lift this barrier. The absolute time scale is not identifiable from a single snapshot of any sample size and requires an external clock. For tumor-immune tissue, the results imply that directed cell-cell interactions are in principle recoverable from one section, that the direction cannot be inferred from the mean densities or a covariance matrix, but from the wavenumber dependence of the cross and autospectra, and that a baseline section determines the interaction structure but not the absolute rates.