{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2005:X4EAEVHOZOXLZOZE4KY3PUPZRJ","short_pith_number":"pith:X4EAEVHO","canonical_record":{"source":{"id":"math/0511460","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2005-11-18T00:29:10Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e2412d6fc383b80893932e8c3be80068c63739977e093cc20bfa48b08eac9d1b","abstract_canon_sha256":"051b0aed13fbf18d739dd5051a1d0b1865a583dfbadba0c94a2599fee1086114"},"schema_version":"1.0"},"canonical_sha256":"bf080254eecbaebcbb24e2b1b7d1f98a5e98898bda1d9250e15fe482d6609869","source":{"kind":"arxiv","id":"math/0511460","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0511460","created_at":"2026-05-18T04:00:02Z"},{"alias_kind":"arxiv_version","alias_value":"math/0511460v1","created_at":"2026-05-18T04:00:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0511460","created_at":"2026-05-18T04:00:02Z"},{"alias_kind":"pith_short_12","alias_value":"X4EAEVHOZOXL","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"X4EAEVHOZOXLZOZE","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"X4EAEVHO","created_at":"2026-05-18T12:25:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2005:X4EAEVHOZOXLZOZE4KY3PUPZRJ","target":"record","payload":{"canonical_record":{"source":{"id":"math/0511460","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2005-11-18T00:29:10Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"e2412d6fc383b80893932e8c3be80068c63739977e093cc20bfa48b08eac9d1b","abstract_canon_sha256":"051b0aed13fbf18d739dd5051a1d0b1865a583dfbadba0c94a2599fee1086114"},"schema_version":"1.0"},"canonical_sha256":"bf080254eecbaebcbb24e2b1b7d1f98a5e98898bda1d9250e15fe482d6609869","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:02.972578Z","signature_b64":"kfDnaWUnl6Ng+wFdMelVWzE9nmzbK0i3cSW855utdZNbFh8014zI+/3IQBVHXj0+PN51XBUaeHtBMlB/xq1hCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf080254eecbaebcbb24e2b1b7d1f98a5e98898bda1d9250e15fe482d6609869","last_reissued_at":"2026-05-18T04:00:02.971884Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:02.971884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0511460","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:00:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+ibagBY7RDErfmILEczRtn1X2kvkhfE/aQ6h/N9MIl6Vdqp9JIo4EQJCChif4lbut/4KqbRNxS69B2DYWvrlDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:54:44.661623Z"},"content_sha256":"b43f409e7cb6bc252af13c05653b585eb75040de2a1e2327fa8dec2360be02ce","schema_version":"1.0","event_id":"sha256:b43f409e7cb6bc252af13c05653b585eb75040de2a1e2327fa8dec2360be02ce"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2005:X4EAEVHOZOXLZOZE4KY3PUPZRJ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Group-theoretic algorithms for matrix multiplication","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.GR","authors_text":"Balazs Szegedy, Christopher Umans, Henry Cohn, Robert Kleinberg","submitted_at":"2005-11-18T00:29:10Z","abstract_excerpt":"We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication exponent less than 3, the asymptotically fastest of which achieves exponent 2.41. We present two conjectures regarding specific improvements, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0511460","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:00:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4odcNUUH0BZ17t2i4Mcb2NHkLaVqf8Ci/GtlyqK9cTQlqPVb2yLzqc5RMSVBSoslxe82jR2lBbwp3b/QpMPpAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T23:54:44.662381Z"},"content_sha256":"8e55053906b5d957464814495f2d47f9b6ccdaa2a2dd6f232b1f4ab7c0a9eaa6","schema_version":"1.0","event_id":"sha256:8e55053906b5d957464814495f2d47f9b6ccdaa2a2dd6f232b1f4ab7c0a9eaa6"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X4EAEVHOZOXLZOZE4KY3PUPZRJ/bundle.json","state_url":"https://pith.science/pith/X4EAEVHOZOXLZOZE4KY3PUPZRJ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X4EAEVHOZOXLZOZE4KY3PUPZRJ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-11T23:54:44Z","links":{"resolver":"https://pith.science/pith/X4EAEVHOZOXLZOZE4KY3PUPZRJ","bundle":"https://pith.science/pith/X4EAEVHOZOXLZOZE4KY3PUPZRJ/bundle.json","state":"https://pith.science/pith/X4EAEVHOZOXLZOZE4KY3PUPZRJ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X4EAEVHOZOXLZOZE4KY3PUPZRJ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:X4EAEVHOZOXLZOZE4KY3PUPZRJ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"051b0aed13fbf18d739dd5051a1d0b1865a583dfbadba0c94a2599fee1086114","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.GR","submitted_at":"2005-11-18T00:29:10Z","title_canon_sha256":"e2412d6fc383b80893932e8c3be80068c63739977e093cc20bfa48b08eac9d1b"},"schema_version":"1.0","source":{"id":"math/0511460","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0511460","created_at":"2026-05-18T04:00:02Z"},{"alias_kind":"arxiv_version","alias_value":"math/0511460v1","created_at":"2026-05-18T04:00:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0511460","created_at":"2026-05-18T04:00:02Z"},{"alias_kind":"pith_short_12","alias_value":"X4EAEVHOZOXL","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"X4EAEVHOZOXLZOZE","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"X4EAEVHO","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:8e55053906b5d957464814495f2d47f9b6ccdaa2a2dd6f232b1f4ab7c0a9eaa6","target":"graph","created_at":"2026-05-18T04:00:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication exponent less than 3, the asymptotically fastest of which achieves exponent 2.41. We present two conjectures regarding specific improvements, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2.","authors_text":"Balazs Szegedy, Christopher Umans, Henry Cohn, Robert Kleinberg","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.GR","submitted_at":"2005-11-18T00:29:10Z","title":"Group-theoretic algorithms for matrix multiplication"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0511460","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:b43f409e7cb6bc252af13c05653b585eb75040de2a1e2327fa8dec2360be02ce","target":"record","created_at":"2026-05-18T04:00:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"051b0aed13fbf18d739dd5051a1d0b1865a583dfbadba0c94a2599fee1086114","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.GR","submitted_at":"2005-11-18T00:29:10Z","title_canon_sha256":"e2412d6fc383b80893932e8c3be80068c63739977e093cc20bfa48b08eac9d1b"},"schema_version":"1.0","source":{"id":"math/0511460","kind":"arxiv","version":1}},"canonical_sha256":"bf080254eecbaebcbb24e2b1b7d1f98a5e98898bda1d9250e15fe482d6609869","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bf080254eecbaebcbb24e2b1b7d1f98a5e98898bda1d9250e15fe482d6609869","first_computed_at":"2026-05-18T04:00:02.971884Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:02.971884Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kfDnaWUnl6Ng+wFdMelVWzE9nmzbK0i3cSW855utdZNbFh8014zI+/3IQBVHXj0+PN51XBUaeHtBMlB/xq1hCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:02.972578Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0511460","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b43f409e7cb6bc252af13c05653b585eb75040de2a1e2327fa8dec2360be02ce","sha256:8e55053906b5d957464814495f2d47f9b6ccdaa2a2dd6f232b1f4ab7c0a9eaa6"],"state_sha256":"0fa7c2adc4f5419a20cd46bf930a4a3e8d7275809bd9546652a374b317fc1215"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OoUvnfjTA1ZxDeX1Mz9f2mX/vnCSfn9VrxTJk0SmLAdRTHkhY6i3yyx6dV/JTWb7I2rFBhxCzhcmIcu55q+EDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T23:54:44.666898Z","bundle_sha256":"abd60dc808a8d466ea86fdc0c2f38937ecc7932d8adb76812d66641868429e99"}}