{
  "abstract": "Making all symmetry-allowed tensor representations computationally accessible remains a central challenge in equivariant atomistic learning. Clebsch--Gordan tensor products (CGTPs) are computationally demanding, whereas efficient $SO(2)$-based alternatives lack a unified treatment of natural- and unnatural-parity representations. We first introduce a generalized Wigner-$6j$ convolution that exactly recouples interactions involving additional node representations, replacing the first edge-level tensor-product intermediate with reusable node features. This reduces edge computation and storage but leaves the $\\mathcal O(L^5)$ angular scaling of the sparse CGTPs unchanged. To address this remaining bottleneck, we develop a local $O(2)$ framework whose convolutions scale as $\\mathcal O(L^3)$. The framework explicitly accounts for both rotations and reflections. Its linear maps, tensor products, and gated nonlinearities, combined with frame transformations, ensure global $O(3)$ equivariance for both spatial parities. Incorporating time-reversal labels extends this construction to $O(3)\\times\\mathbb Z_2^{\\mathcal T}$. We apply the framework to magnetic TACE (mTACE), using distinct interactions for systems with and without spin--orbit coupling. On collinear CrN and noncollinear Fe benchmarks, mTACE achieves substantial reductions in atomic- and magnetic-force errors. We also provide EquivariantX, a library for global and local $O(2)$-equivariant computation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.16592",
  "author_total": 2,
  "authors": [
    {
      "name": "Zemin Xu"
    },
    {
      "name": "Wenbo Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/cvmx3ehwnz6f6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.16592",
  "fulltext_url": null,
  "id": "cvmx3ehwnz6f6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.16592",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.mtrl-sci"
    ],
    [
      "arxiv",
      "physics.chem-ph"
    ],
    [
      "openalex",
      "2505|Machine Learning in Materials Science"
    ]
  ],
  "openalex": "W7203706166",
  "pdf_url": "https://arxiv.org/pdf/2608.16592",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/cvmx3ehwnz6f6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:52Z",
      "native_id": "2608.16592",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.16592"
    }
  ],
  "subjects": [
    {
      "field": "chemistry",
      "origin": "arxiv:physics.chem-ph",
      "subfield": "physical"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.mtrl-sci",
      "subfield": "condensed-matter"
    }
  ],
  "title": "Through the Looking-Glass: Efficient Parity-Complete Learning via Local $O(2)$ Frames",
  "title_text": "Through the Looking-Glass: Efficient Parity-Complete Learning via Local O(2) Frames",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:52Z",
  "url": "https://research.draconitic.com/paper/cvmx3ehwnz6f6",
  "venue": "arXiv",
  "versions": []
}