geometry-topology / Invariants of point configurations
Pozdnyakov, Willatt, Bartók, Ortner, Csányi and Ceriotti showed in 2020 that the 2-, 3- and 4-point correlations of an atomic neighbour density are incomplete: noncongruent environments can share them exactly. Every degeneracy found since was dissolved by going to a higher correlation order, leaving open whether the trispectrum (5-body correlations), or any fixed finite order, separates all noncongruent environments. It does not. There are noncongruent three-dimensional environments agreeing on every cluster of up to seven neighbours, and for each finite correlation order and angular cutoff there are continuous families of noncongruent environments with identical retained features.
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Append-only history
The key ingredients were already in the literature, decades old and in distant fields; what was missing was anyone connecting them to this question.
Research memory
Pozdnyakov, Willatt, Bartók, Ortner, Csányi and Ceriotti showed in 2020 that the 2-, 3- and 4-point correlations of an atomic neighbour density are incomplete: noncongruent environments can share them exactly. Every degeneracy found since was dissolved by going to a higher correlation order, leaving open whether the trispectrum (5-body correlations), or any fixed finite order, separates all noncongruent environments. It does not. There are noncongruent three-dimensional environments agreeing on every cluster of up to seven neighbours, and for each finite correlation order and angular cutoff there are continuous families of noncongruent environments with identical retained features.
The key ingredients were already in the literature, decades old and in distant fields; what was missing was anyone connecting them to this question.
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