Completeness of Fixed-Order Atom-Centered Descriptors

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.

Record

Source: arXiv:2607.26984 - Using large language models to probe the limits of atom-centered structural descriptors
VibeMathed
Added: Aug 17, 2026
Links: Pozdnyakov et al., the 2020 paper that posed it arxiv.org

Comments

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construction · #1
Michelangelo Domina and Michele Ceriotti, using Claude Opus 4.8 and Fable, Codex 5.5 and Sol 5.6Aug 17, 2026
That credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.
AI involvement
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— a person and a model developed the result together.
$The paper is as much about the usage pattern as the result, and says so in its title. The authors had been stuck on these questions for years and could not get past the known Pozdnyakov examples. Coding agents built on Claude (Opus 4.8 and Fable) and Codex (5.5 and Sol 5.6) were given a summary of the field's literature and, across unstructured iteration, produced the degenerate configurations by locating results in unrelated communities and recognising what they implied here, among them the homometric structures studied in musical rhythm theory. The authors verified the constructions manually and with model-generated code, and lifted the cyclic degeneracies to three dimensions themselves. They also ran a reproducibility experiment, eight runs per model under each of two prompt conditions, to test how reliably a query of this kind lands.$

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