  In this paper, we address the challenge of obtaining a comprehensive and
symmetric representation of point particle groups, such as atoms in a molecule,
which is crucial in physics and theoretical chemistry. The problem has become
even more important with the widespread adoption of machine-learning techniques
in science, as it underpins the capacity of models to accurately reproduce
physical relationships while being consistent with fundamental symmetries and
conservation laws. However, some of the descriptors that are commonly used to
represent point clouds -- most notably those based on discretized correlations
of the neighbor density, that underpin most of the existing ML models of matter
at the atomic scale -- are unable to distinguish between special arrangements
of particles in three dimensions. This makes it impossible to machine learn
their properties. Atom-density correlations are provably complete in the limit
in which they simultaneously describe the mutual relationship between all
atoms, which is impractical. We present a novel approach to construct
descriptors of \emph{finite} correlations based on the relative arrangement of
particle triplets, which can be employed to create symmetry-adapted models with
universal approximation capabilities, which have the resolution of the neighbor
discretization as the sole convergence parameter. Our strategy is demonstrated
on a class of atomic arrangements that are specifically built to defy a broad
class of conventional symmetric descriptors, showcasing its potential for
addressing their limitations.
