{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:4N5WCWMPH723TC6N77ZKCQQ5E3","short_pith_number":"pith:4N5WCWMP","canonical_record":{"source":{"id":"1608.07486","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-08-26T15:06:53Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"5dd8126cce583f1b328834024d0d81ec15a6d920e4533132784d29db285b65e5","abstract_canon_sha256":"a3585e812998cba256cb47f4cf0c58521819e5e0e4efe94731d360c615b0869d"},"schema_version":"1.0"},"canonical_sha256":"e37b61598f3ff5b98bcdfff2a1421d26e6e82f8a77d2a17a61d83df4639df50a","source":{"kind":"arxiv","id":"1608.07486","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.07486","created_at":"2026-05-18T01:07:52Z"},{"alias_kind":"arxiv_version","alias_value":"1608.07486v1","created_at":"2026-05-18T01:07:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.07486","created_at":"2026-05-18T01:07:52Z"},{"alias_kind":"pith_short_12","alias_value":"4N5WCWMPH723","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4N5WCWMPH723TC6N","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4N5WCWMP","created_at":"2026-05-18T12:29:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:4N5WCWMPH723TC6N77ZKCQQ5E3","target":"record","payload":{"canonical_record":{"source":{"id":"1608.07486","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-08-26T15:06:53Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"5dd8126cce583f1b328834024d0d81ec15a6d920e4533132784d29db285b65e5","abstract_canon_sha256":"a3585e812998cba256cb47f4cf0c58521819e5e0e4efe94731d360c615b0869d"},"schema_version":"1.0"},"canonical_sha256":"e37b61598f3ff5b98bcdfff2a1421d26e6e82f8a77d2a17a61d83df4639df50a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:52.912055Z","signature_b64":"4W3XLhasA7X9uhG0HfROQV1lhsdfCuZf5+juQvGWKuzPJnPJluTseEIzRGp/l3ImjNDlu8I7ShToDha3w0HWBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e37b61598f3ff5b98bcdfff2a1421d26e6e82f8a77d2a17a61d83df4639df50a","last_reissued_at":"2026-05-18T01:07:52.911646Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:52.911646Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.07486","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":"2026-05-18T01:07:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YvYTd+nwRhGWT+nJ9M/GFFCA2FLgW9wE2vfp5IEuxW5e1n93B8+ieYZKQl8NhvZA0KqM4EOQrPBMxDASQ2DWCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:03:37.351120Z"},"content_sha256":"56d89d0f72c1f6eef0f9820c91e96a0074e84e96b924817d83ae48344ad4ebb9","schema_version":"1.0","event_id":"sha256:56d89d0f72c1f6eef0f9820c91e96a0074e84e96b924817d83ae48344ad4ebb9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:4N5WCWMPH723TC6N77ZKCQQ5E3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A $2n^2-log(n)-1$ lower bound for the border rank of matrix multiplication","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"cs.CC","authors_text":"J.M. Landsberg, Mateusz Micha{\\l}ek","submitted_at":"2016-08-26T15:06:53Z","abstract_excerpt":"Let M_n denote the matrix multiplication tensor for nxn matrices. We use the border substitution method combined with Koszul flattenings to prove the border rank lower bound of 2n^2-log(n)-1 for M_n."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.07486","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":"2026-05-18T01:07:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OCB3xVXjwe2u6KPPacaeZVZef3fo+eyLj3L4rcAX+/36ICrRQgsfJ6v/dyQgaQzn6GZsVwMcxQPScLRLmlWxCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-11T06:03:37.352124Z"},"content_sha256":"ec174dcb88fab86989ca6672429dd30db18194900aede5f2c19ee663b28ca70f","schema_version":"1.0","event_id":"sha256:ec174dcb88fab86989ca6672429dd30db18194900aede5f2c19ee663b28ca70f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/4N5WCWMPH723TC6N77ZKCQQ5E3/bundle.json","state_url":"https://pith.science/pith/4N5WCWMPH723TC6N77ZKCQQ5E3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/4N5WCWMPH723TC6N77ZKCQQ5E3/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":"2026-06-11T06:03:37Z","links":{"resolver":"https://pith.science/pith/4N5WCWMPH723TC6N77ZKCQQ5E3","bundle":"https://pith.science/pith/4N5WCWMPH723TC6N77ZKCQQ5E3/bundle.json","state":"https://pith.science/pith/4N5WCWMPH723TC6N77ZKCQQ5E3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/4N5WCWMPH723TC6N77ZKCQQ5E3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:4N5WCWMPH723TC6N77ZKCQQ5E3","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":"a3585e812998cba256cb47f4cf0c58521819e5e0e4efe94731d360c615b0869d","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-08-26T15:06:53Z","title_canon_sha256":"5dd8126cce583f1b328834024d0d81ec15a6d920e4533132784d29db285b65e5"},"schema_version":"1.0","source":{"id":"1608.07486","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.07486","created_at":"2026-05-18T01:07:52Z"},{"alias_kind":"arxiv_version","alias_value":"1608.07486v1","created_at":"2026-05-18T01:07:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.07486","created_at":"2026-05-18T01:07:52Z"},{"alias_kind":"pith_short_12","alias_value":"4N5WCWMPH723","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_16","alias_value":"4N5WCWMPH723TC6N","created_at":"2026-05-18T12:29:58Z"},{"alias_kind":"pith_short_8","alias_value":"4N5WCWMP","created_at":"2026-05-18T12:29:58Z"}],"graph_snapshots":[{"event_id":"sha256:ec174dcb88fab86989ca6672429dd30db18194900aede5f2c19ee663b28ca70f","target":"graph","created_at":"2026-05-18T01:07:52Z","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":"Let M_n denote the matrix multiplication tensor for nxn matrices. We use the border substitution method combined with Koszul flattenings to prove the border rank lower bound of 2n^2-log(n)-1 for M_n.","authors_text":"J.M. Landsberg, Mateusz Micha{\\l}ek","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-08-26T15:06:53Z","title":"A $2n^2-log(n)-1$ lower bound for the border rank of matrix multiplication"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.07486","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":"sha256:56d89d0f72c1f6eef0f9820c91e96a0074e84e96b924817d83ae48344ad4ebb9","target":"record","created_at":"2026-05-18T01:07:52Z","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":"a3585e812998cba256cb47f4cf0c58521819e5e0e4efe94731d360c615b0869d","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2016-08-26T15:06:53Z","title_canon_sha256":"5dd8126cce583f1b328834024d0d81ec15a6d920e4533132784d29db285b65e5"},"schema_version":"1.0","source":{"id":"1608.07486","kind":"arxiv","version":1}},"canonical_sha256":"e37b61598f3ff5b98bcdfff2a1421d26e6e82f8a77d2a17a61d83df4639df50a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e37b61598f3ff5b98bcdfff2a1421d26e6e82f8a77d2a17a61d83df4639df50a","first_computed_at":"2026-05-18T01:07:52.911646Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:07:52.911646Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4W3XLhasA7X9uhG0HfROQV1lhsdfCuZf5+juQvGWKuzPJnPJluTseEIzRGp/l3ImjNDlu8I7ShToDha3w0HWBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:07:52.912055Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.07486","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:56d89d0f72c1f6eef0f9820c91e96a0074e84e96b924817d83ae48344ad4ebb9","sha256:ec174dcb88fab86989ca6672429dd30db18194900aede5f2c19ee663b28ca70f"],"state_sha256":"3dffe522e4b9abf99f7dfd92581604a770498f7c9cf4cadd488bec49c61b6c41"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+M+8404+Gm5292qCEzg5R30EO1zyhQKJ6hFiy0sj9fTGzN86CdKSZDDHXk9apEIDG4w77FIZPvORZAOjuqbNDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-11T06:03:37.356495Z","bundle_sha256":"729cf8d8f5708364251eca1c6766c480db549dd83bf9ae0c52c4161e1bf1f6e5"}}