{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:DBSOTRIKCWKEWP4PFEKACKJJ5W","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":"486f2c959a357c7cd8db869c30151cff45410dd56e1269a0d0f0ce8bafe37e08","cross_cats_sorted":["cs.CC","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-23T19:55:05Z","title_canon_sha256":"b0ad8a4cc527ac0f3b8b0a4315bb2c72bd89137ca52b9ec2ca08dbf120e917d4"},"schema_version":"1.0","source":{"id":"1609.07476","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1609.07476","created_at":"2026-07-05T00:03:11Z"},{"alias_kind":"arxiv_version","alias_value":"1609.07476v2","created_at":"2026-07-05T00:03:11Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.07476","created_at":"2026-07-05T00:03:11Z"},{"alias_kind":"pith_short_12","alias_value":"DBSOTRIKCWKE","created_at":"2026-07-05T00:03:11Z"},{"alias_kind":"pith_short_16","alias_value":"DBSOTRIKCWKEWP4P","created_at":"2026-07-05T00:03:11Z"},{"alias_kind":"pith_short_8","alias_value":"DBSOTRIK","created_at":"2026-07-05T00:03:11Z"}],"graph_snapshots":[{"event_id":"sha256:cd39c3cceebddd909d50259bad1b14c5c736549e397d0cc3d81ff5928b1aa026","target":"graph","created_at":"2026-07-05T00:03:11Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1609.07476/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We present an upper bound on the exponent of the asymptotic behaviour of the tensor rank of a family of tensors defined by the complete graph on $k$ vertices. For $k\\geq4$, we show that the exponent per edge is at most 0.77, outperforming the best known upper bound on the exponent per edge for matrix multiplication ($k=3$), which is approximately 0.79. We raise the question whether for some $k$ the exponent per edge can be below $2/3$, i.e. can outperform matrix multiplication even if the matrix multiplication exponent equals 2. In order to obtain our results, we generalise to higher order ten","authors_text":"Jeroen Zuiddam, Matthias Christandl, P\\'eter Vrana","cross_cats":["cs.CC","quant-ph"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-23T19:55:05Z","title":"Asymptotic tensor rank of graph tensors: beyond matrix multiplication"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.07476","kind":"arxiv","version":2},"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:c842171176f224e2a1a4b63ab1278c8ec036bdd5c1bbb1178bb10a2d102caaaa","target":"record","created_at":"2026-07-05T00:03:11Z","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":"486f2c959a357c7cd8db869c30151cff45410dd56e1269a0d0f0ce8bafe37e08","cross_cats_sorted":["cs.CC","quant-ph"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-09-23T19:55:05Z","title_canon_sha256":"b0ad8a4cc527ac0f3b8b0a4315bb2c72bd89137ca52b9ec2ca08dbf120e917d4"},"schema_version":"1.0","source":{"id":"1609.07476","kind":"arxiv","version":2}},"canonical_sha256":"1864e9c50a15944b3f8f2914012929ed95182a2dcbee518e705553f56da950e7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1864e9c50a15944b3f8f2914012929ed95182a2dcbee518e705553f56da950e7","first_computed_at":"2026-07-05T00:03:11.371334Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:03:11.371334Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NDYk0cf9EWwrb8Yd5yotFfi6I91nag3owQX6I4wPfFGFFndHENk+gCheltkAT1WpyQiKSsrnLZMPENMBBQVdBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T00:03:11.371699Z","signed_message":"canonical_sha256_bytes"},"source_id":"1609.07476","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c842171176f224e2a1a4b63ab1278c8ec036bdd5c1bbb1178bb10a2d102caaaa","sha256:cd39c3cceebddd909d50259bad1b14c5c736549e397d0cc3d81ff5928b1aa026"],"state_sha256":"2ac059e2043699364b2f7bfa644ca5b2f6fbfb13acb260dff44d32d55c8e2594"}