Polyatomic Complexes: A topologically-informed learning representation for atomistic systems

Abstract

A representation of a molecule or material should be invariant to the symmetries of physics, unique, continuous, efficient and general. These properties, however, are hard to satisfy at once: a descriptor invariant under the full orthogonal group $O(3)$ gives a molecule and its mirror image the same value, and so cannot distinguish enantiomers whose properties differ. Pozdnyakov showed this follows from the invariance itself, not from a lack of parameters. We show the criteria can be met at once if the geometric map is graded by the sign character of $O(3)$ and pooled multisymmetrically. We construct such a map $\hat{I}¦$: its even block factors through the Gram matrix and is provably chirality-blind, while its parity-odd block of signed triple products separates enantiomers on an open dense full-measure set of interacting configurations. Two standard obstructions to uniqueness, fixed output length and componentwise pooling, are artifacts of the pooling rule, removed by multisymmetric power sums of order at most $N$. We establish uniqueness for a complete descriptor $\hat{I}{¦}^{\star }$ built from the distance matrix, signed volumes and atom types, injective up to $SE(3)\times S_N$ on all configurations. $\hat{I}{¦}^{\star }$ is non-constructive, however; the implemented map is the bounded-cutoff $\hat{I}¦$, generically injective, pooling at order $2$, running in $O(N^2)$, or $O(N)$ with neighbor lists. The algebraic core is machine-checked in Lean 4. Because the underlying object is a cell complex, it also yields invariant, stable topological features $\hat{I}¨$ (persistent homology and a Hodge-Laplacian spectrum) encoding global ring and cage structure invisible to bounded-cutoff descriptors. The pair $(\hat{I}¦,\hat{I}¨)$ feeds a compact parity-graded equivariant transformer.

- revised draft