{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:AZUN6RDNGNAC3PZLZ2E5GCHPTF","short_pith_number":"pith:AZUN6RDN","canonical_record":{"source":{"id":"1104.5097","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-04-27T09:36:31Z","cross_cats_sorted":["cs.SC","math.CO"],"title_canon_sha256":"e9b3b37d3b4187d3270150ce5ddf7b005eb29c5f9653b14144573d5bcad72229","abstract_canon_sha256":"0e212ee1627585d1ab11155773b6bf32c2957740281322300be63c2a4072d970"},"schema_version":"1.0"},"canonical_sha256":"0668df446d33402dbf2bce89d308ef9958908ba579b8f39e5ec75983250453ef","source":{"kind":"arxiv","id":"1104.5097","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.5097","created_at":"2026-05-18T03:56:35Z"},{"alias_kind":"arxiv_version","alias_value":"1104.5097v2","created_at":"2026-05-18T03:56:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.5097","created_at":"2026-05-18T03:56:35Z"},{"alias_kind":"pith_short_12","alias_value":"AZUN6RDNGNAC","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"AZUN6RDNGNAC3PZL","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"AZUN6RDN","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:AZUN6RDNGNAC3PZLZ2E5GCHPTF","target":"record","payload":{"canonical_record":{"source":{"id":"1104.5097","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-04-27T09:36:31Z","cross_cats_sorted":["cs.SC","math.CO"],"title_canon_sha256":"e9b3b37d3b4187d3270150ce5ddf7b005eb29c5f9653b14144573d5bcad72229","abstract_canon_sha256":"0e212ee1627585d1ab11155773b6bf32c2957740281322300be63c2a4072d970"},"schema_version":"1.0"},"canonical_sha256":"0668df446d33402dbf2bce89d308ef9958908ba579b8f39e5ec75983250453ef","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:56:35.413110Z","signature_b64":"F7rtKB7R2cOEytb+tKZ54ysEuMQoML2BSvmnsPLvrYTPWh7pMV/Z0pvTK7tyTGOxgZDw1MgTmWyezLpvmcl3Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0668df446d33402dbf2bce89d308ef9958908ba579b8f39e5ec75983250453ef","last_reissued_at":"2026-05-18T03:56:35.412697Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:56:35.412697Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1104.5097","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:56:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yx7KzotUqjNT8AUTg53YyTYIw6WBB5sDnbuS+wfv5rCMyyBPb3Rtk4LABWGdbwTrrCEacfDi1xVeGSyFqUqIAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T20:17:38.089579Z"},"content_sha256":"88756073d17a3d76027391d7ec1b5d4ea20a0aacbab1ad7b61a35136864f453a","schema_version":"1.0","event_id":"sha256:88756073d17a3d76027391d7ec1b5d4ea20a0aacbab1ad7b61a35136864f453a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:AZUN6RDNGNAC3PZLZ2E5GCHPTF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Search and test algorithms for Triple Product Property triples","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.SC","math.CO"],"primary_cat":"math.GR","authors_text":"Ivo Hedtke, Sandeep Murthy","submitted_at":"2011-04-27T09:36:31Z","abstract_excerpt":"In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group $G$ satisfying the Triple Product Property (TPP) as a means to bound the exponent $\\omega$ of matrix multiplication. We present two new characterizations of the TPP, which are useful for theoretical considerations and for TPP test algorithms. With this we describe all known TPP tests and implement them in GAP algorithms. We also compare their runtime. Furthermore we show that the search for subgroup TPP triples of nontrivial size in a nonabelian group can b"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.5097","kind":"arxiv","version":2},"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":"2026-05-18T03:56:35Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a90ziVFilM0E6vD9p3Ej8GMzYt0bXGjgqIB/E+0l8X8LaiX0qVLGwDi2MrNV410lDMzVAt57MOpI7xdy2RpuAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T20:17:38.090199Z"},"content_sha256":"e1072cc31bfbd8aaaf80e4d27e27e4f18a5d93d149c255f8279b66ab985d426c","schema_version":"1.0","event_id":"sha256:e1072cc31bfbd8aaaf80e4d27e27e4f18a5d93d149c255f8279b66ab985d426c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/AZUN6RDNGNAC3PZLZ2E5GCHPTF/bundle.json","state_url":"https://pith.science/pith/AZUN6RDNGNAC3PZLZ2E5GCHPTF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/AZUN6RDNGNAC3PZLZ2E5GCHPTF/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":"2026-05-31T20:17:38Z","links":{"resolver":"https://pith.science/pith/AZUN6RDNGNAC3PZLZ2E5GCHPTF","bundle":"https://pith.science/pith/AZUN6RDNGNAC3PZLZ2E5GCHPTF/bundle.json","state":"https://pith.science/pith/AZUN6RDNGNAC3PZLZ2E5GCHPTF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/AZUN6RDNGNAC3PZLZ2E5GCHPTF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:AZUN6RDNGNAC3PZLZ2E5GCHPTF","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":"0e212ee1627585d1ab11155773b6bf32c2957740281322300be63c2a4072d970","cross_cats_sorted":["cs.SC","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-04-27T09:36:31Z","title_canon_sha256":"e9b3b37d3b4187d3270150ce5ddf7b005eb29c5f9653b14144573d5bcad72229"},"schema_version":"1.0","source":{"id":"1104.5097","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.5097","created_at":"2026-05-18T03:56:35Z"},{"alias_kind":"arxiv_version","alias_value":"1104.5097v2","created_at":"2026-05-18T03:56:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.5097","created_at":"2026-05-18T03:56:35Z"},{"alias_kind":"pith_short_12","alias_value":"AZUN6RDNGNAC","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"AZUN6RDNGNAC3PZL","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"AZUN6RDN","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:e1072cc31bfbd8aaaf80e4d27e27e4f18a5d93d149c255f8279b66ab985d426c","target":"graph","created_at":"2026-05-18T03:56:35Z","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":"In 2003 COHN and UMANS introduced a group-theoretic approach to fast matrix multiplication. This involves finding large subsets of a group $G$ satisfying the Triple Product Property (TPP) as a means to bound the exponent $\\omega$ of matrix multiplication. We present two new characterizations of the TPP, which are useful for theoretical considerations and for TPP test algorithms. With this we describe all known TPP tests and implement them in GAP algorithms. We also compare their runtime. Furthermore we show that the search for subgroup TPP triples of nontrivial size in a nonabelian group can b","authors_text":"Ivo Hedtke, Sandeep Murthy","cross_cats":["cs.SC","math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-04-27T09:36:31Z","title":"Search and test algorithms for Triple Product Property triples"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.5097","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":"sha256:88756073d17a3d76027391d7ec1b5d4ea20a0aacbab1ad7b61a35136864f453a","target":"record","created_at":"2026-05-18T03:56:35Z","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":"0e212ee1627585d1ab11155773b6bf32c2957740281322300be63c2a4072d970","cross_cats_sorted":["cs.SC","math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-04-27T09:36:31Z","title_canon_sha256":"e9b3b37d3b4187d3270150ce5ddf7b005eb29c5f9653b14144573d5bcad72229"},"schema_version":"1.0","source":{"id":"1104.5097","kind":"arxiv","version":2}},"canonical_sha256":"0668df446d33402dbf2bce89d308ef9958908ba579b8f39e5ec75983250453ef","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0668df446d33402dbf2bce89d308ef9958908ba579b8f39e5ec75983250453ef","first_computed_at":"2026-05-18T03:56:35.412697Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:56:35.412697Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"F7rtKB7R2cOEytb+tKZ54ysEuMQoML2BSvmnsPLvrYTPWh7pMV/Z0pvTK7tyTGOxgZDw1MgTmWyezLpvmcl3Cw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:56:35.413110Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.5097","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:88756073d17a3d76027391d7ec1b5d4ea20a0aacbab1ad7b61a35136864f453a","sha256:e1072cc31bfbd8aaaf80e4d27e27e4f18a5d93d149c255f8279b66ab985d426c"],"state_sha256":"5e7e23d981826ea484ddeee0f33238c968a5fd8ee4f64ddecdffa50f8760cca0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ThqT+0FM/REYU+MnE6NwS//9XFt21Y/8WDEs0g1QFMbiKwdoikdOwJFaLPz1i8WXvI1L5rSg8Z6DsU1cu0L0DA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T20:17:38.093624Z","bundle_sha256":"a6ca61e3a74feb53d1ca1e94dc11e90618d83b344072b5f3493033822cb37cea"}}