Completeness of Fixed-Order Atom-Centered Descriptors
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Statement
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.
Context
The key ingredients were already in the literature, decades old and in distant fields; what was missing was anyone connecting them to this question.
A six-year-old completeness question from the atomistic machine-learning literature, settled negatively at every fixed order. The authors themselves note the degenerate structures are far from plausible chemistry and largely reinforce a view the field had already reached.
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The paper is as much about the usage pattern as the result, and says so in its title. The authors had been stuck on these questions for years and could not get past the known Pozdnyakov examples. Coding agents built on Claude (Opus 4.8 and Fable) and Codex (5.5 and Sol 5.6) were given a summary of the field's literature and, across unstructured iteration, produced the degenerate configurations by locating results in unrelated communities and recognising what they implied here, among them the homometric structures studied in musical rhythm theory. The authors verified the constructions manually and with model-generated code, and lifted the cyclic degeneracies to three dimensions themselves. They also ran a reproducibility experiment, eight runs per model under each of two prompt conditions, to test how reliably a query of this kind lands.
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