{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:1992:HMDWWH2M2DTCFSYPETUR6EGEAG","short_pith_number":"pith:HMDWWH2M","canonical_record":{"source":{"id":"math/9207206","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1992-07-06T16:04:20Z","cross_cats_sorted":[],"title_canon_sha256":"3bf31c253bff72e135353f86d2945fffb3a2a86c8652298a0f8dbdc49dd3d144","abstract_canon_sha256":"37edf3eaa9b970d2cf3b5c351328f214d2cb4debedef443ae2049c24da6b0f03"},"schema_version":"1.0"},"canonical_sha256":"3b076b1f4cd0e622cb0f24e91f10c401b1fef0f3d88bac2b5c665dee75e5f1e4","source":{"kind":"arxiv","id":"math/9207206","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9207206","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"arxiv_version","alias_value":"math/9207206v1","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9207206","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"pith_short_12","alias_value":"HMDWWH2M2DTC","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"HMDWWH2M2DTCFSYP","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"HMDWWH2M","created_at":"2026-05-18T12:25:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:1992:HMDWWH2M2DTCFSYPETUR6EGEAG","target":"record","payload":{"canonical_record":{"source":{"id":"math/9207206","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.FA","submitted_at":"1992-07-06T16:04:20Z","cross_cats_sorted":[],"title_canon_sha256":"3bf31c253bff72e135353f86d2945fffb3a2a86c8652298a0f8dbdc49dd3d144","abstract_canon_sha256":"37edf3eaa9b970d2cf3b5c351328f214d2cb4debedef443ae2049c24da6b0f03"},"schema_version":"1.0"},"canonical_sha256":"3b076b1f4cd0e622cb0f24e91f10c401b1fef0f3d88bac2b5c665dee75e5f1e4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:52.744310Z","signature_b64":"26bwZOBXnB8plldMaQVIp0Bx8buH0+Z/To+1UX97moVNi79tsOiHn9N7DfPvpg3SmxCJ6rbYoomuVhbt8Ce6Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b076b1f4cd0e622cb0f24e91f10c401b1fef0f3d88bac2b5c665dee75e5f1e4","last_reissued_at":"2026-05-18T01:05:52.743797Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:52.743797Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/9207206","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":"2026-05-18T01:05:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"29ivPWbq4pwsh9SAqmll918I4qCuv79DjZxoVOY8DB7AqB0pDVUsU+s8cnQjBfI1cLDV3ByT3IISCJvHrw8ZBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T13:51:14.287932Z"},"content_sha256":"affa468e0147abea11d95a98c0facb497347bc44dbedf7af3e437e37f4bff047","schema_version":"1.0","event_id":"sha256:affa468e0147abea11d95a98c0facb497347bc44dbedf7af3e437e37f4bff047"},{"event_type":"graph_snapshot","subject_pith_number":"pith:1992:HMDWWH2M2DTCFSYPETUR6EGEAG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Banach Spaces Of The Type Of Tsirelson","license":"","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Irene Deliyanni, Spiros A. Argyros","submitted_at":"1992-07-06T16:04:20Z","abstract_excerpt":"To any pair ( M , theta ) where M is a family of finite subsets of N compact in the pointwise topology, and 0<theta < 1 , we associate a Tsirelson-type Banach space T_M^theta . It is shown that if the Cantor-Bendixson index of M is greater than n and theta >{1/n} then T_M^theta is reflexive. Moreover, if the Cantor-Bendixson index of M is greater than omega then T_M^theta does not contain any l^p, while if the Cantor-Bendixson index of M is finite thenT_M^theta contains some l^p or c_o . In particular, if M ={ A subset N : |A| leq n } and {1/n}<theta <1 then T_M^theta is isomorphic to some l^p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9207206","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":"2026-05-18T01:05:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"T8rtFbf1lwI8RiGrkITBxkls0I95/ynOkfLJt6NrJdzUU8kMIbmteFHJ2lVXBGc7BLBFyL2lG2Rtt0YM5r3xAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T13:51:14.288272Z"},"content_sha256":"9b8e704112197955d4f75895ee6b99f0ecb20826c77ee707e92365cf699700cc","schema_version":"1.0","event_id":"sha256:9b8e704112197955d4f75895ee6b99f0ecb20826c77ee707e92365cf699700cc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HMDWWH2M2DTCFSYPETUR6EGEAG/bundle.json","state_url":"https://pith.science/pith/HMDWWH2M2DTCFSYPETUR6EGEAG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HMDWWH2M2DTCFSYPETUR6EGEAG/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":"2026-06-26T13:51:14Z","links":{"resolver":"https://pith.science/pith/HMDWWH2M2DTCFSYPETUR6EGEAG","bundle":"https://pith.science/pith/HMDWWH2M2DTCFSYPETUR6EGEAG/bundle.json","state":"https://pith.science/pith/HMDWWH2M2DTCFSYPETUR6EGEAG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HMDWWH2M2DTCFSYPETUR6EGEAG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:1992:HMDWWH2M2DTCFSYPETUR6EGEAG","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":"37edf3eaa9b970d2cf3b5c351328f214d2cb4debedef443ae2049c24da6b0f03","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1992-07-06T16:04:20Z","title_canon_sha256":"3bf31c253bff72e135353f86d2945fffb3a2a86c8652298a0f8dbdc49dd3d144"},"schema_version":"1.0","source":{"id":"math/9207206","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/9207206","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"arxiv_version","alias_value":"math/9207206v1","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/9207206","created_at":"2026-05-18T01:05:52Z"},{"alias_kind":"pith_short_12","alias_value":"HMDWWH2M2DTC","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_16","alias_value":"HMDWWH2M2DTCFSYP","created_at":"2026-05-18T12:25:47Z"},{"alias_kind":"pith_short_8","alias_value":"HMDWWH2M","created_at":"2026-05-18T12:25:47Z"}],"graph_snapshots":[{"event_id":"sha256:9b8e704112197955d4f75895ee6b99f0ecb20826c77ee707e92365cf699700cc","target":"graph","created_at":"2026-05-18T01:05:52Z","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":"To any pair ( M , theta ) where M is a family of finite subsets of N compact in the pointwise topology, and 0<theta < 1 , we associate a Tsirelson-type Banach space T_M^theta . It is shown that if the Cantor-Bendixson index of M is greater than n and theta >{1/n} then T_M^theta is reflexive. Moreover, if the Cantor-Bendixson index of M is greater than omega then T_M^theta does not contain any l^p, while if the Cantor-Bendixson index of M is finite thenT_M^theta contains some l^p or c_o . In particular, if M ={ A subset N : |A| leq n } and {1/n}<theta <1 then T_M^theta is isomorphic to some l^p","authors_text":"Irene Deliyanni, Spiros A. Argyros","cross_cats":[],"headline":"","license":"","primary_cat":"math.FA","submitted_at":"1992-07-06T16:04:20Z","title":"Banach Spaces Of The Type Of Tsirelson"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/9207206","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:affa468e0147abea11d95a98c0facb497347bc44dbedf7af3e437e37f4bff047","target":"record","created_at":"2026-05-18T01:05:52Z","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":"37edf3eaa9b970d2cf3b5c351328f214d2cb4debedef443ae2049c24da6b0f03","cross_cats_sorted":[],"license":"","primary_cat":"math.FA","submitted_at":"1992-07-06T16:04:20Z","title_canon_sha256":"3bf31c253bff72e135353f86d2945fffb3a2a86c8652298a0f8dbdc49dd3d144"},"schema_version":"1.0","source":{"id":"math/9207206","kind":"arxiv","version":1}},"canonical_sha256":"3b076b1f4cd0e622cb0f24e91f10c401b1fef0f3d88bac2b5c665dee75e5f1e4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"3b076b1f4cd0e622cb0f24e91f10c401b1fef0f3d88bac2b5c665dee75e5f1e4","first_computed_at":"2026-05-18T01:05:52.743797Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:05:52.743797Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"26bwZOBXnB8plldMaQVIp0Bx8buH0+Z/To+1UX97moVNi79tsOiHn9N7DfPvpg3SmxCJ6rbYoomuVhbt8Ce6Dw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:05:52.744310Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/9207206","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:affa468e0147abea11d95a98c0facb497347bc44dbedf7af3e437e37f4bff047","sha256:9b8e704112197955d4f75895ee6b99f0ecb20826c77ee707e92365cf699700cc"],"state_sha256":"2764edc7f790faf9e561faa5870d810ecc71ef03a44fba9b3d8cda82094c8286"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"O+ciElA9pMHyQ/y1x0w8KY0vQPoBEIZ1IPGyimcEqE2O+fuV9awRFG7DDR/VTpcJO8VVgNf5MzeSOQevwo/nDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T13:51:14.290200Z","bundle_sha256":"a67d7b0d614649c3eb27bd5c096c6736f9f2294c390f08fe40b1d493755b188a"}}