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Mapping the angular integral to a structured Gauss--Legendre and Fourier tensor-product grid decouples the radial channel contractions from the angular transforms. The antisymmetric parity-odd Clebsch--Gordan channels, unreachable by the symmetric pointwise product on a scalar $S^2$ grid, are recovered through the surface-curl pairing $\\hat r \\cdot [\\nabla_{S^2} A \\times \\nabla_{S^2} B]$, the spher"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.15073","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"physics.comp-ph","submitted_at":"2026-05-14T16:59:00Z","cross_cats_sorted":["cond-mat.mtrl-sci","physics.chem-ph"],"title_canon_sha256":"b0caeb9aa9cad73e92ee8c217bcad18fb2542d9c1d2ec4ed312e123f8f4eddde","abstract_canon_sha256":"7a242647d6f4b7d650096015e4c5c6a83787a9c779bde28124a74f1377b9d1f3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.2","canonical_sha256":"ef0e3014166c895e82a41680367286489085e6560f68ba3dac95e55188660f86","last_reissued_at":"2026-05-17T21:57:19.328196Z","signature_status":"unsigned_v0","first_computed_at":"2026-05-17T21:40:26.020099Z"},"graph_snapshot":{"paper":{"title":"Fast contracted Clebsch--Gordan tensor products for equivariant graph neural networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"An O(L^3) algorithm evaluates contracted Clebsch-Gordan tensor products for O(3)-equivariant machine learning potentials using a structured grid and surface-curl pairing.","cross_cats":["cond-mat.mtrl-sci","physics.chem-ph"],"primary_cat":"physics.comp-ph","authors_text":"Anton Bochkarev, Ralf Drautz, Yury Lysogorskiy","submitted_at":"2026-05-14T16:59:00Z","abstract_excerpt":"We present an $\\mathcal{O}(L^3)$ algorithm for evaluating contracted Clebsch--Gordan tensor products in $\\mathrm{O}(3)$-equivariant machine learning potentials at fixed Canonical Polyadic (CP) rank. 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