{"paper":{"title":"Grokking Finite-Dimensional Algebra","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Neural networks grok algebra multiplication once they recover the bilinear product from the structure tensor.","cross_cats":["cs.AI","math.RA"],"primary_cat":"cs.LG","authors_text":"Guillaume Dumas, Guillaume Rabusseau, Pascal Jr Tikeng Notsawo","submitted_at":"2026-02-23T05:55:52Z","abstract_excerpt":"This paper investigates the grokking phenomenon, which refers to the sudden transition from a long memorization to generalization observed during neural networks training, in the context of learning multiplication in finite-dimensional algebras (FDA). While prior work on grokking has focused mainly on group operations, we extend the analysis to more general algebraic structures, including non-associative, non-commutative, and non-unital algebras. We show that learning group operations is a special case of learning FDA, and that learning multiplication in FDA amounts to learning a bilinear prod"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Learning multiplication in finite-dimensional algebras amounts to learning a bilinear product specified by the algebra's structure tensor, and grokking emerges naturally as models learn discrete representations for algebras over finite fields.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That the experimental models are actually learning the algebra's multiplication via the structure tensor rather than some other shortcut that happens to correlate with the target operation.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Neural networks learning multiplication in finite-dimensional algebras show grokking whose timing depends on algebraic properties like commutativity and the rank/sparsity of the structure tensor.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Neural networks grok algebra multiplication once they recover the bilinear product from the structure tensor.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"4707ba5ff95829468742a284349410ef5f18f7b6da3bcf3d27b40ebcf879d145"},"source":{"id":"2602.19533","kind":"arxiv","version":2},"verdict":{"id":"89ef4905-57d0-48a7-b620-e0b095c76281","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T20:19:51.927536Z","strongest_claim":"Learning multiplication in finite-dimensional algebras amounts to learning a bilinear product specified by the algebra's structure tensor, and grokking emerges naturally as models learn discrete representations for algebras over finite fields.","one_line_summary":"Neural networks learning multiplication in finite-dimensional algebras show grokking whose timing depends on algebraic properties like commutativity and the rank/sparsity of the structure tensor.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That the experimental models are actually learning the algebra's multiplication via the structure tensor rather than some other shortcut that happens to correlate with the target operation.","pith_extraction_headline":"Neural networks grok algebra multiplication once they recover the bilinear product from the structure tensor."},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"bd224ddaa0b5a28cbea69a2887510e6e0aabfdf7a83c74ab7c9ac0b4c12da1f8"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}