{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:T4HD7RGAMDXLFP6YCG5VGHKUYY","short_pith_number":"pith:T4HD7RGA","schema_version":"1.0","canonical_sha256":"9f0e3fc4c060eeb2bfd811bb531d54c60b5fd151f776a6801ae661123109ddf4","source":{"kind":"arxiv","id":"2602.19533","version":2},"attestation_state":"computed","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"},"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":false,"formal_links_present":true},"canonical_record":{"source":{"id":"2602.19533","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.LG","submitted_at":"2026-02-23T05:55:52Z","cross_cats_sorted":["cs.AI","math.RA"],"title_canon_sha256":"feed14295d552fef19e547aaa5c0248c6852af8b588d8875faa02f59a6fabbe1","abstract_canon_sha256":"58560f6e7a9c2e2cca50ad7d25a1547a43029aaa41cc9c5723add94232ffe862"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:39:15.998563Z","signature_b64":"VSkwPdMR81k0kyP+O0ZTwdQlo6bsGw8NcXAJDb0wk6bbDzz+UNT8+5+48MFAc466pNep8DD28Uj1w460NrbnDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9f0e3fc4c060eeb2bfd811bb531d54c60b5fd151f776a6801ae661123109ddf4","last_reissued_at":"2026-05-17T23:39:15.997675Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:39:15.997675Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2602.19533","created_at":"2026-05-17T23:39:15.997796+00:00"},{"alias_kind":"arxiv_version","alias_value":"2602.19533v2","created_at":"2026-05-17T23:39:15.997796+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2602.19533","created_at":"2026-05-17T23:39:15.997796+00:00"},{"alias_kind":"pith_short_12","alias_value":"T4HD7RGAMDXL","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_16","alias_value":"T4HD7RGAMDXLFP6Y","created_at":"2026-05-18T12:33:37.589309+00:00"},{"alias_kind":"pith_short_8","alias_value":"T4HD7RGA","created_at":"2026-05-18T12:33:37.589309+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":2,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/T4HD7RGAMDXLFP6YCG5VGHKUYY","json":"https://pith.science/pith/T4HD7RGAMDXLFP6YCG5VGHKUYY.json","graph_json":"https://pith.science/api/pith-number/T4HD7RGAMDXLFP6YCG5VGHKUYY/graph.json","events_json":"https://pith.science/api/pith-number/T4HD7RGAMDXLFP6YCG5VGHKUYY/events.json","paper":"https://pith.science/paper/T4HD7RGA"},"agent_actions":{"view_html":"https://pith.science/pith/T4HD7RGAMDXLFP6YCG5VGHKUYY","download_json":"https://pith.science/pith/T4HD7RGAMDXLFP6YCG5VGHKUYY.json","view_paper":"https://pith.science/paper/T4HD7RGA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2602.19533&json=true","fetch_graph":"https://pith.science/api/pith-number/T4HD7RGAMDXLFP6YCG5VGHKUYY/graph.json","fetch_events":"https://pith.science/api/pith-number/T4HD7RGAMDXLFP6YCG5VGHKUYY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T4HD7RGAMDXLFP6YCG5VGHKUYY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T4HD7RGAMDXLFP6YCG5VGHKUYY/action/storage_attestation","attest_author":"https://pith.science/pith/T4HD7RGAMDXLFP6YCG5VGHKUYY/action/author_attestation","sign_citation":"https://pith.science/pith/T4HD7RGAMDXLFP6YCG5VGHKUYY/action/citation_signature","submit_replication":"https://pith.science/pith/T4HD7RGAMDXLFP6YCG5VGHKUYY/action/replication_record"}},"created_at":"2026-05-17T23:39:15.997796+00:00","updated_at":"2026-05-17T23:39:15.997796+00:00"}