{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:24EXCCKAJDGYZ6DG5A7L24GILW","short_pith_number":"pith:24EXCCKA","canonical_record":{"source":{"id":"1503.03387","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-03-11T15:50:41Z","cross_cats_sorted":[],"title_canon_sha256":"aa9bbdf8e3b1ebbc3822db30156898618dca7aa93a02b1778fe0340f1d351df6","abstract_canon_sha256":"4257969e7a50ab29582a85589da6af777762dfbba776bdd6bb04220fc9bdd08d"},"schema_version":"1.0"},"canonical_sha256":"d70971094048cd8cf866e83ebd70c85d9d74431bbef86cd5cb2b329ea046bf57","source":{"kind":"arxiv","id":"1503.03387","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.03387","created_at":"2026-05-18T02:25:03Z"},{"alias_kind":"arxiv_version","alias_value":"1503.03387v1","created_at":"2026-05-18T02:25:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.03387","created_at":"2026-05-18T02:25:03Z"},{"alias_kind":"pith_short_12","alias_value":"24EXCCKAJDGY","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"24EXCCKAJDGYZ6DG","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"24EXCCKA","created_at":"2026-05-18T12:28:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:24EXCCKAJDGYZ6DG5A7L24GILW","target":"record","payload":{"canonical_record":{"source":{"id":"1503.03387","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-03-11T15:50:41Z","cross_cats_sorted":[],"title_canon_sha256":"aa9bbdf8e3b1ebbc3822db30156898618dca7aa93a02b1778fe0340f1d351df6","abstract_canon_sha256":"4257969e7a50ab29582a85589da6af777762dfbba776bdd6bb04220fc9bdd08d"},"schema_version":"1.0"},"canonical_sha256":"d70971094048cd8cf866e83ebd70c85d9d74431bbef86cd5cb2b329ea046bf57","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:03.718874Z","signature_b64":"1uLXmv4I4GpPZmAmSvMzyScFIruLdF+RWVvKhjanB/XcEvPAyBWlf7jukLLxdo7zEppC1co2A8w0iWRnRiNJDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d70971094048cd8cf866e83ebd70c85d9d74431bbef86cd5cb2b329ea046bf57","last_reissued_at":"2026-05-18T02:25:03.718469Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:03.718469Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1503.03387","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":"2026-05-18T02:25:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SZwlOhWZJAxTy67aNDgFMXwPEXJGgyuD5CzTucV3MY2wFIyGTBSjlT3ICMHAqFhAuNblCfwjBKL/zIe/oar+CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T11:25:31.111071Z"},"content_sha256":"df3da219d69286fb02a548b707dd649ca7601ea786f8d8a65e37907341eb2185","schema_version":"1.0","event_id":"sha256:df3da219d69286fb02a548b707dd649ca7601ea786f8d8a65e37907341eb2185"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:24EXCCKAJDGYZ6DG5A7L24GILW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Levels of generalized expansiveness","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Jie Li, Ruifeng Zhang","submitted_at":"2015-03-11T15:50:41Z","abstract_excerpt":"We study a class of generalized expansive dynamical systems for which at most countable orbits can be accompanied by an arbitrary given orbit. Examples of different levels of generalized expansiveness are constructed. When the dynamical system is countable, a characterization of n-expansiveness is given for any natural number n, and as a consequence examples of dynamical systems with van der Waerden depth equal to any given countable ordinal are demonstrated, which solves open questions existing in the literature."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.03387","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":"2026-05-18T02:25:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kDEU1T2WLnNYAXTvoW1HsXt5nFaImYnyM4R8DYoJh0JBCpVccSFk0w9ABRPyny6NOQUyv2DFuxCJkotPBDm2Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T11:25:31.111432Z"},"content_sha256":"f714919c5bb5b46e21aac85e9f98d912cadcc58ba963b55a9b5ec702803d7a23","schema_version":"1.0","event_id":"sha256:f714919c5bb5b46e21aac85e9f98d912cadcc58ba963b55a9b5ec702803d7a23"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/24EXCCKAJDGYZ6DG5A7L24GILW/bundle.json","state_url":"https://pith.science/pith/24EXCCKAJDGYZ6DG5A7L24GILW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/24EXCCKAJDGYZ6DG5A7L24GILW/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":"2026-06-01T11:25:31Z","links":{"resolver":"https://pith.science/pith/24EXCCKAJDGYZ6DG5A7L24GILW","bundle":"https://pith.science/pith/24EXCCKAJDGYZ6DG5A7L24GILW/bundle.json","state":"https://pith.science/pith/24EXCCKAJDGYZ6DG5A7L24GILW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/24EXCCKAJDGYZ6DG5A7L24GILW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:24EXCCKAJDGYZ6DG5A7L24GILW","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":"4257969e7a50ab29582a85589da6af777762dfbba776bdd6bb04220fc9bdd08d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-03-11T15:50:41Z","title_canon_sha256":"aa9bbdf8e3b1ebbc3822db30156898618dca7aa93a02b1778fe0340f1d351df6"},"schema_version":"1.0","source":{"id":"1503.03387","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.03387","created_at":"2026-05-18T02:25:03Z"},{"alias_kind":"arxiv_version","alias_value":"1503.03387v1","created_at":"2026-05-18T02:25:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.03387","created_at":"2026-05-18T02:25:03Z"},{"alias_kind":"pith_short_12","alias_value":"24EXCCKAJDGY","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"24EXCCKAJDGYZ6DG","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"24EXCCKA","created_at":"2026-05-18T12:28:59Z"}],"graph_snapshots":[{"event_id":"sha256:f714919c5bb5b46e21aac85e9f98d912cadcc58ba963b55a9b5ec702803d7a23","target":"graph","created_at":"2026-05-18T02:25:03Z","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":"We study a class of generalized expansive dynamical systems for which at most countable orbits can be accompanied by an arbitrary given orbit. Examples of different levels of generalized expansiveness are constructed. When the dynamical system is countable, a characterization of n-expansiveness is given for any natural number n, and as a consequence examples of dynamical systems with van der Waerden depth equal to any given countable ordinal are demonstrated, which solves open questions existing in the literature.","authors_text":"Jie Li, Ruifeng Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-03-11T15:50:41Z","title":"Levels of generalized expansiveness"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.03387","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:df3da219d69286fb02a548b707dd649ca7601ea786f8d8a65e37907341eb2185","target":"record","created_at":"2026-05-18T02:25:03Z","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":"4257969e7a50ab29582a85589da6af777762dfbba776bdd6bb04220fc9bdd08d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2015-03-11T15:50:41Z","title_canon_sha256":"aa9bbdf8e3b1ebbc3822db30156898618dca7aa93a02b1778fe0340f1d351df6"},"schema_version":"1.0","source":{"id":"1503.03387","kind":"arxiv","version":1}},"canonical_sha256":"d70971094048cd8cf866e83ebd70c85d9d74431bbef86cd5cb2b329ea046bf57","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d70971094048cd8cf866e83ebd70c85d9d74431bbef86cd5cb2b329ea046bf57","first_computed_at":"2026-05-18T02:25:03.718469Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:25:03.718469Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1uLXmv4I4GpPZmAmSvMzyScFIruLdF+RWVvKhjanB/XcEvPAyBWlf7jukLLxdo7zEppC1co2A8w0iWRnRiNJDw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:25:03.718874Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.03387","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:df3da219d69286fb02a548b707dd649ca7601ea786f8d8a65e37907341eb2185","sha256:f714919c5bb5b46e21aac85e9f98d912cadcc58ba963b55a9b5ec702803d7a23"],"state_sha256":"9fcfd1090cc2406b6c3b9c96c0ca2f93848f3be8847bcf7a3108d21904aa6ef2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EPvYHcIIPaqRd3Trx2yRrlmBe+PrKO9+3xnvgo/JYOUaPD/yWrmGEtB/TYcjn9jq7Q8xGw9pV0NMUv9sJAprBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T11:25:31.113662Z","bundle_sha256":"a8555be71701f549e184f0b11badb57d779809a8744871aabdf386b8cf94126b"}}