{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:BZVFLJD6GONTPSXIF4KNCIVJNS","short_pith_number":"pith:BZVFLJD6","canonical_record":{"source":{"id":"2105.08394","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-05-18T09:46:31Z","cross_cats_sorted":[],"title_canon_sha256":"60e106a5b744a772c7d257efd26e46696c5fc2c2e584a93d0c600630e98a1a96","abstract_canon_sha256":"54d5c516fda6088a3efc25546def45ad3b8dc865b19a4c183134ef593883f08e"},"schema_version":"1.0"},"canonical_sha256":"0e6a55a47e339b37cae82f14d122a96c87994baafa2bc791fea3a5fdfd9228c9","source":{"kind":"arxiv","id":"2105.08394","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2105.08394","created_at":"2026-07-05T03:05:02Z"},{"alias_kind":"arxiv_version","alias_value":"2105.08394v2","created_at":"2026-07-05T03:05:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.08394","created_at":"2026-07-05T03:05:02Z"},{"alias_kind":"pith_short_12","alias_value":"BZVFLJD6GONT","created_at":"2026-07-05T03:05:02Z"},{"alias_kind":"pith_short_16","alias_value":"BZVFLJD6GONTPSXI","created_at":"2026-07-05T03:05:02Z"},{"alias_kind":"pith_short_8","alias_value":"BZVFLJD6","created_at":"2026-07-05T03:05:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:BZVFLJD6GONTPSXIF4KNCIVJNS","target":"record","payload":{"canonical_record":{"source":{"id":"2105.08394","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-05-18T09:46:31Z","cross_cats_sorted":[],"title_canon_sha256":"60e106a5b744a772c7d257efd26e46696c5fc2c2e584a93d0c600630e98a1a96","abstract_canon_sha256":"54d5c516fda6088a3efc25546def45ad3b8dc865b19a4c183134ef593883f08e"},"schema_version":"1.0"},"canonical_sha256":"0e6a55a47e339b37cae82f14d122a96c87994baafa2bc791fea3a5fdfd9228c9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:05:02.468905Z","signature_b64":"ohJCnvs+4Jc65wTiw9QNdd13g0NrKghj5IiFYsSwtoz57Q3Zjc+fjUoWkbICmJeeBGXxHYJH+CG2rkuEqVdWDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0e6a55a47e339b37cae82f14d122a96c87994baafa2bc791fea3a5fdfd9228c9","last_reissued_at":"2026-07-05T03:05:02.468527Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:05:02.468527Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2105.08394","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":"2026-07-05T03:05:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Y2BbTXqkN0lJ27PWmofJb6eVcrNZPY97OKYE34a3PwWOdQPY7xb9GU7B89J8R9QaC+QlPZ1dqHI3eVsYmBtcCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T14:09:57.545991Z"},"content_sha256":"f1c49827e99fa616acbe358369ea4046c71a7a30c52d39629ce79e91773b0077","schema_version":"1.0","event_id":"sha256:f1c49827e99fa616acbe358369ea4046c71a7a30c52d39629ce79e91773b0077"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:BZVFLJD6GONTPSXIF4KNCIVJNS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The slice rank of a direct sum","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"W. T. Gowers","submitted_at":"2021-05-18T09:46:31Z","abstract_excerpt":"We show that the slice rank of the direct sum of two tensors is equal to the sum of their slice ranks. The upper bound is trivial, but the lower bound needs more than a one-line proof, for reasons we explain. This result generalizes the fact, shown by Tao, that the slice rank of a diagonal tensor is equal to the number of non-zero entries of that tensor."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.08394","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2105.08394/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T03:05:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VEn9kP8le6V5ZiyZmsydvTf6AeRlO8/ohf6r61XwGZMfov/mf+CABIZy9SxRdImVv693BxhvgUFX5n2b5yrvBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T14:09:57.549918Z"},"content_sha256":"aab52050386f3a880e7d43a9e07013a01bed973d7ecde6369b5d4c4d105a205e","schema_version":"1.0","event_id":"sha256:aab52050386f3a880e7d43a9e07013a01bed973d7ecde6369b5d4c4d105a205e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BZVFLJD6GONTPSXIF4KNCIVJNS/bundle.json","state_url":"https://pith.science/pith/BZVFLJD6GONTPSXIF4KNCIVJNS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BZVFLJD6GONTPSXIF4KNCIVJNS/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":"2026-08-13T14:09:57Z","links":{"resolver":"https://pith.science/pith/BZVFLJD6GONTPSXIF4KNCIVJNS","bundle":"https://pith.science/pith/BZVFLJD6GONTPSXIF4KNCIVJNS/bundle.json","state":"https://pith.science/pith/BZVFLJD6GONTPSXIF4KNCIVJNS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BZVFLJD6GONTPSXIF4KNCIVJNS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:BZVFLJD6GONTPSXIF4KNCIVJNS","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":"54d5c516fda6088a3efc25546def45ad3b8dc865b19a4c183134ef593883f08e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-05-18T09:46:31Z","title_canon_sha256":"60e106a5b744a772c7d257efd26e46696c5fc2c2e584a93d0c600630e98a1a96"},"schema_version":"1.0","source":{"id":"2105.08394","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2105.08394","created_at":"2026-07-05T03:05:02Z"},{"alias_kind":"arxiv_version","alias_value":"2105.08394v2","created_at":"2026-07-05T03:05:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2105.08394","created_at":"2026-07-05T03:05:02Z"},{"alias_kind":"pith_short_12","alias_value":"BZVFLJD6GONT","created_at":"2026-07-05T03:05:02Z"},{"alias_kind":"pith_short_16","alias_value":"BZVFLJD6GONTPSXI","created_at":"2026-07-05T03:05:02Z"},{"alias_kind":"pith_short_8","alias_value":"BZVFLJD6","created_at":"2026-07-05T03:05:02Z"}],"graph_snapshots":[{"event_id":"sha256:aab52050386f3a880e7d43a9e07013a01bed973d7ecde6369b5d4c4d105a205e","target":"graph","created_at":"2026-07-05T03:05:02Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2105.08394/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We show that the slice rank of the direct sum of two tensors is equal to the sum of their slice ranks. The upper bound is trivial, but the lower bound needs more than a one-line proof, for reasons we explain. This result generalizes the fact, shown by Tao, that the slice rank of a diagonal tensor is equal to the number of non-zero entries of that tensor.","authors_text":"W. T. Gowers","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-05-18T09:46:31Z","title":"The slice rank of a direct sum"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2105.08394","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:f1c49827e99fa616acbe358369ea4046c71a7a30c52d39629ce79e91773b0077","target":"record","created_at":"2026-07-05T03:05:02Z","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":"54d5c516fda6088a3efc25546def45ad3b8dc865b19a4c183134ef593883f08e","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2021-05-18T09:46:31Z","title_canon_sha256":"60e106a5b744a772c7d257efd26e46696c5fc2c2e584a93d0c600630e98a1a96"},"schema_version":"1.0","source":{"id":"2105.08394","kind":"arxiv","version":2}},"canonical_sha256":"0e6a55a47e339b37cae82f14d122a96c87994baafa2bc791fea3a5fdfd9228c9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0e6a55a47e339b37cae82f14d122a96c87994baafa2bc791fea3a5fdfd9228c9","first_computed_at":"2026-07-05T03:05:02.468527Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:05:02.468527Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ohJCnvs+4Jc65wTiw9QNdd13g0NrKghj5IiFYsSwtoz57Q3Zjc+fjUoWkbICmJeeBGXxHYJH+CG2rkuEqVdWDw==","signature_status":"signed_v1","signed_at":"2026-07-05T03:05:02.468905Z","signed_message":"canonical_sha256_bytes"},"source_id":"2105.08394","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f1c49827e99fa616acbe358369ea4046c71a7a30c52d39629ce79e91773b0077","sha256:aab52050386f3a880e7d43a9e07013a01bed973d7ecde6369b5d4c4d105a205e"],"state_sha256":"7f24d3eed402091f03cdbbfc75eae1769274097e43649aaf7662802fefe639b0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"znlPbKUPt2p6dJKwLDIe3N1fc5SwEiKgoDg0wSpgEn+lFE8OprVXPOezQ+cC+pNj3z+DtTmNzZ1vel3MJi9+AA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T14:09:57.555323Z","bundle_sha256":"a34a1808bc7e9bb53844e1dbe8ec606c991fdc384f28dfcf22d4b2f15f1fdf0a"}}