{"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. 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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We present an O(L^3) algorithm for evaluating contracted Clebsch--Gordan tensor products in O(3)-equivariant machine learning potentials at fixed Canonical Polyadic (CP) rank.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The structured Gauss-Legendre and Fourier grid combined with the surface-curl pairing accurately recovers all required Clebsch-Gordan channels, including antisymmetric parity-odd ones, without introducing numerical errors or breaking equivariance for the target L range.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"An O(L^3) algorithm computes contracted Clebsch-Gordan tensor products for equivariant ML potentials using a structured angular grid and spherical Poisson bracket to handle parity-odd terms at fixed CP rank.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"86119939ed771527f3de2282b9d56bb2c7e57a1e5f60e57793418b063d6f630e"},"source":{"id":"2605.15073","kind":"arxiv","version":1},"verdict":{"id":"c6a135a0-83a5-4e68-8532-27c08f416c93","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:59:51.677938Z","strongest_claim":"We present an O(L^3) algorithm for evaluating contracted Clebsch--Gordan tensor products in O(3)-equivariant machine learning potentials at fixed Canonical Polyadic (CP) rank.","one_line_summary":"An O(L^3) algorithm computes contracted Clebsch-Gordan tensor products for equivariant ML potentials using a structured angular grid and spherical Poisson bracket to handle parity-odd terms at fixed CP rank.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The structured Gauss-Legendre and Fourier grid combined with the surface-curl pairing accurately recovers all required Clebsch-Gordan channels, including antisymmetric parity-odd ones, without introducing numerical errors or breaking equivariance for the target L range.","pith_extraction_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."},"references":{"count":55,"sample":[{"doi":"","year":null,"title":"Parity-odd identity recovery For each parity-odd triple (l1, l2, l) we evaluate Eq.(15) on every (m 1, m2, m) entry withm=m 1 +m 2 and ex- tractκ −1(l1, l2, l) = integral/G lm l1m1 l2m2 from the non- ","work_id":"538eb5e9-570d-4808-ba09-e4f50127ebcc","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Across all parity-even triples up to (4,4,8), the maximum integral was∼10 −10, set by the quadrature","work_id":"4cf6fe80-af32-49b3-9d55-481836ac79d6","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Antisymmetry under1↔2exchange The surface-curl integrand ˆr·[∇ S2 Yl1m1 × ∇S2 Yl2m2] flips sign exactly under exchange of the two factors. Di- rect evaluation across all parity-odd triples up to (4,4,","work_id":"9d5917cf-74e3-4df9-8797-f14bff6a3d37","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"End-to-end consistency on random synthetic fields To verify that the surface-curl construction reproduces the contracted CG output not only on plain spheri- cal harmonics but on arbitrary input fields","work_id":"d084d23f-e9ba-46a6-90d6-9bcc5f1f0783","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Φ(sym/anti) lm = X n1,n2 M (sym/anti) n1n2 × X m1m2 Glm l1m1 l2m2 An1l1m1 An2l2m2 14 withA nlm random Gaussian,l 1 =l 2,M sym symmetric, M anti antisymmetric","work_id":"e9a0098e-5f7f-4262-888f-bd0b9d9e6258","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":55,"snapshot_sha256":"f3f619d05778459dda3e3649a6d36e2988310e2989526a73c99475a14f770859","internal_anchors":5},"formal_canon":{"evidence_count":1,"snapshot_sha256":"51e472c90ed5aa09548504700cf2fe1c2aee3f002466bcac42774e541c38b836"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}