Circular Topological Transforms: Geometric Chirality with Completeness

Geometric graphs such as point clouds, 3D shapes, and molecular conformations require representations that are both (1) *$\mathrm{SE}(n)$-complete*, that representations agree if and only if they differ only by translations and rotations, and (2) *chirality-aware*, that distinct geometries and their mirror images remain distinguishable. Existing approaches do not provide both properties at scale. $\mathrm{SE}(n)$-invariant geometric networks are limited by geometric Weisfeiler–Leman expressivity and capture chirality only locally, while popular topological directional transforms, such as the persistent homology transform (PHT), are injective (can distinguish any shape), but therefore not $\mathrm{SE}(n)$-complete. In this work, we propose *Circular Topological Transforms* (CIRCUiT${}^2$), a principled methodology that turns admissible injective directional transforms such as the PHT to be both $\mathrm{SE}(n)$-complete and chirality-aware on shape classes admitting an equivariant frame. Our construction organizes directional information into ordered circular scans: proper rotations are absorbed by changes of frame, whereas reflections induce a deterministic reversal of the scan. We also show that two naive approaches (group averaging and sorting canonicalization) to symmetrizing directional transforms will fail to detect chirality. Applying our methodology to the PHT, $\mathrm{C}\mathrm{IRC}$-$\mathrm{PHT}$ achieves perfect chirality-sensitive retrieval on multiple molecular benchmarks and captures long-range chirality beyond the interaction radius of the evaluated equivariant deep neural network models.