{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:ZMLF4TVGPOIVZ6FTAQLFQQ7IJ3","short_pith_number":"pith:ZMLF4TVG","canonical_record":{"source":{"id":"1604.07875","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-26T22:21:45Z","cross_cats_sorted":[],"title_canon_sha256":"590e2834dc235fdaa6428a53a7d342e652df888bbd14d03ec4cdd880e6cc09d0","abstract_canon_sha256":"4817501aaf39ca0a3e173cfdf1b88100ff5c19640bb721385fb2395e74af6670"},"schema_version":"1.0"},"canonical_sha256":"cb165e4ea67b915cf8b304165843e84edecbe2858c9e74f45c954a0bde11c42a","source":{"kind":"arxiv","id":"1604.07875","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.07875","created_at":"2026-05-18T00:43:09Z"},{"alias_kind":"arxiv_version","alias_value":"1604.07875v2","created_at":"2026-05-18T00:43:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.07875","created_at":"2026-05-18T00:43:09Z"},{"alias_kind":"pith_short_12","alias_value":"ZMLF4TVGPOIV","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"ZMLF4TVGPOIVZ6FT","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"ZMLF4TVG","created_at":"2026-05-18T12:30:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:ZMLF4TVGPOIVZ6FTAQLFQQ7IJ3","target":"record","payload":{"canonical_record":{"source":{"id":"1604.07875","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-26T22:21:45Z","cross_cats_sorted":[],"title_canon_sha256":"590e2834dc235fdaa6428a53a7d342e652df888bbd14d03ec4cdd880e6cc09d0","abstract_canon_sha256":"4817501aaf39ca0a3e173cfdf1b88100ff5c19640bb721385fb2395e74af6670"},"schema_version":"1.0"},"canonical_sha256":"cb165e4ea67b915cf8b304165843e84edecbe2858c9e74f45c954a0bde11c42a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:09.455337Z","signature_b64":"IKXph7fx7WDsmjGz2niLcWdn3J3Vc6+YiZxSpwXYutZIYUylQmQieYSPNcPtx4Q6Jur6fZiR3LhQLYzj7FpPBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb165e4ea67b915cf8b304165843e84edecbe2858c9e74f45c954a0bde11c42a","last_reissued_at":"2026-05-18T00:43:09.454657Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:09.454657Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1604.07875","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":"2026-05-18T00:43:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GgqctJOlngsYAere0bptF/U6SyFQwGICrAJ3T8SfQTfSptRLgJP3zJr6LxydE/uTHTZVHVlqzlkxeZTO9B3ADg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T08:58:01.861923Z"},"content_sha256":"6088ec9d7046af530c5aba9c66b21b15e84b92c0bdeb77ef3d39cdcf88465577","schema_version":"1.0","event_id":"sha256:6088ec9d7046af530c5aba9c66b21b15e84b92c0bdeb77ef3d39cdcf88465577"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:ZMLF4TVGPOIVZ6FTAQLFQQ7IJ3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Szlenk index of injective tensor products and convex hulls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Ryan M Causey","submitted_at":"2016-04-26T22:21:45Z","abstract_excerpt":"Given any Banach space $X$ and any weak*-compact subset $K$ of $X^*$, we compute the Szlenk index of the weak*-closed, convex hull of $K$ as a function of the Szlenk index of $K$. Also as an application, we compute the Szlenk index of any injective tensor product of two operators. In particular, we compute the Szlenk index of an injective tensor product $\\tens$ in terms of $Sz(X)$ and $Sz(Y)$. As another application, we give a complete characterization of those ordinals which occur as the Szlenk index of a Banach space, as well as those ordinals which occur as the Bourgain $\\ell_1$ or $c_0$ in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07875","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":"2026-05-18T00:43:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lvF8B00fbhz76bb8qp2UQIcXLFobYmkVOatoV/8pYZ+VHnlIL1GwOkxubyx5i1DQzX54cL4Oj2wYyXQ5ea2xBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T08:58:01.862659Z"},"content_sha256":"799667cca93077e5e504e4b8ad9c8b0e44613f27b1c61e6e8c2a4bb46255bd6d","schema_version":"1.0","event_id":"sha256:799667cca93077e5e504e4b8ad9c8b0e44613f27b1c61e6e8c2a4bb46255bd6d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZMLF4TVGPOIVZ6FTAQLFQQ7IJ3/bundle.json","state_url":"https://pith.science/pith/ZMLF4TVGPOIVZ6FTAQLFQQ7IJ3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZMLF4TVGPOIVZ6FTAQLFQQ7IJ3/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":"2026-05-23T08:58:01Z","links":{"resolver":"https://pith.science/pith/ZMLF4TVGPOIVZ6FTAQLFQQ7IJ3","bundle":"https://pith.science/pith/ZMLF4TVGPOIVZ6FTAQLFQQ7IJ3/bundle.json","state":"https://pith.science/pith/ZMLF4TVGPOIVZ6FTAQLFQQ7IJ3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZMLF4TVGPOIVZ6FTAQLFQQ7IJ3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:ZMLF4TVGPOIVZ6FTAQLFQQ7IJ3","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":"4817501aaf39ca0a3e173cfdf1b88100ff5c19640bb721385fb2395e74af6670","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-26T22:21:45Z","title_canon_sha256":"590e2834dc235fdaa6428a53a7d342e652df888bbd14d03ec4cdd880e6cc09d0"},"schema_version":"1.0","source":{"id":"1604.07875","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1604.07875","created_at":"2026-05-18T00:43:09Z"},{"alias_kind":"arxiv_version","alias_value":"1604.07875v2","created_at":"2026-05-18T00:43:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1604.07875","created_at":"2026-05-18T00:43:09Z"},{"alias_kind":"pith_short_12","alias_value":"ZMLF4TVGPOIV","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_16","alias_value":"ZMLF4TVGPOIVZ6FT","created_at":"2026-05-18T12:30:53Z"},{"alias_kind":"pith_short_8","alias_value":"ZMLF4TVG","created_at":"2026-05-18T12:30:53Z"}],"graph_snapshots":[{"event_id":"sha256:799667cca93077e5e504e4b8ad9c8b0e44613f27b1c61e6e8c2a4bb46255bd6d","target":"graph","created_at":"2026-05-18T00:43:09Z","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":"Given any Banach space $X$ and any weak*-compact subset $K$ of $X^*$, we compute the Szlenk index of the weak*-closed, convex hull of $K$ as a function of the Szlenk index of $K$. Also as an application, we compute the Szlenk index of any injective tensor product of two operators. In particular, we compute the Szlenk index of an injective tensor product $\\tens$ in terms of $Sz(X)$ and $Sz(Y)$. As another application, we give a complete characterization of those ordinals which occur as the Szlenk index of a Banach space, as well as those ordinals which occur as the Bourgain $\\ell_1$ or $c_0$ in","authors_text":"Ryan M Causey","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-26T22:21:45Z","title":"The Szlenk index of injective tensor products and convex hulls"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07875","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":"sha256:6088ec9d7046af530c5aba9c66b21b15e84b92c0bdeb77ef3d39cdcf88465577","target":"record","created_at":"2026-05-18T00:43:09Z","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":"4817501aaf39ca0a3e173cfdf1b88100ff5c19640bb721385fb2395e74af6670","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2016-04-26T22:21:45Z","title_canon_sha256":"590e2834dc235fdaa6428a53a7d342e652df888bbd14d03ec4cdd880e6cc09d0"},"schema_version":"1.0","source":{"id":"1604.07875","kind":"arxiv","version":2}},"canonical_sha256":"cb165e4ea67b915cf8b304165843e84edecbe2858c9e74f45c954a0bde11c42a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cb165e4ea67b915cf8b304165843e84edecbe2858c9e74f45c954a0bde11c42a","first_computed_at":"2026-05-18T00:43:09.454657Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:43:09.454657Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IKXph7fx7WDsmjGz2niLcWdn3J3Vc6+YiZxSpwXYutZIYUylQmQieYSPNcPtx4Q6Jur6fZiR3LhQLYzj7FpPBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:43:09.455337Z","signed_message":"canonical_sha256_bytes"},"source_id":"1604.07875","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6088ec9d7046af530c5aba9c66b21b15e84b92c0bdeb77ef3d39cdcf88465577","sha256:799667cca93077e5e504e4b8ad9c8b0e44613f27b1c61e6e8c2a4bb46255bd6d"],"state_sha256":"5b3e4c32dcde250db0a303e56357bf2b7307850875f88a0f28fbf69a948094c3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qiSy3dpr9dw5vfZ0pTDS9c0HcmN6UUG8fJM1lEEg8XDsBlh8qyxtOj4Wgew15LFMe/elPUsYtaXKl0cfLIMBCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T08:58:01.866419Z","bundle_sha256":"ce312ce5a699a80ca70fbe0f1a6edc97163c0ae9f655713e9c6cd42586ea6827"}}