{
  "abstract": "Symmetry determines which physical responses of molecules and crystals are allowed and underlies many models used throughout the physical sciences. Yet several widely used machine-learned models learn symmetry only approximately from data, raising the question of whether approximate symmetry is sufficient for scientific prediction. Using the Orb-v3 force field, we show that it is often not: the relaxed space group of a crystal depends on the orientation of the input for $47\\%$ of a $150$-crystal subsample, and relaxation lowers the space group for half of all $399$ crystals studied. To characterize, quantify, and correct such failures, we introduce a tensor algebra, $\\star_G$, defined by convolution over a finite symmetry group and applicable to tensors with noncommuting entries. We prove that the symmetry group determines the algebra's block-diagonal transform and use this structure to analyze model errors on symmetric targets. The resulting projections decompose the average squared error into orthogonal components arising from selection-rule violations, inconsistencies across symmetry-related inputs, and residual error. The removable components reveal how much error arises from broken symmetry, provide reference-free lower bounds on prediction error, and can be eliminated without sacrificing accuracy. For operators, projection followed by blockwise truncation yields the optimal symmetric low-rank approximation. Applying these projections to force predictions reduces both rates of space-group change to at most $2\\%$. Exact selection rules, reference-free error bounds, and symmetry-preserving corrections are therefore available for any trained model without retraining.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.20440",
  "author_total": 8,
  "authors": [
    {
      "name": "Paulina Hoyos"
    },
    {
      "name": "Shashanka Ubaru"
    },
    {
      "name": "Dongsung Huh"
    },
    {
      "name": "Vasileios Kalantzis"
    },
    {
      "name": "Kenneth L. Clarkson"
    },
    {
      "name": "Misha Kilmer"
    },
    {
      "name": "Haim Avron"
    },
    {
      "name": "Lior Horesh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3jddsqf3sg693.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.20440",
  "fulltext_url": null,
  "id": "3jddsqf3sg693",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.20440",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.AI"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "2505|Machine Learning in Materials Science"
    ]
  ],
  "openalex": "W7162020587",
  "pdf_url": "https://arxiv.org/pdf/2605.20440",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3jddsqf3sg693.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2605.20440",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.20440"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Exact Symmetry as Algebra: Group-Structured Tensors that Enforce Physical Selection Rules",
  "title_text": "Exact Symmetry as Algebra: Group-Structured Tensors that Enforce Physical Selection Rules",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/3jddsqf3sg693",
  "venue": "arXiv",
  "versions": []
}