{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2020:VWYUJQTE4LHALF5YGWTUMKHQUF","short_pith_number":"pith:VWYUJQTE","canonical_record":{"source":{"id":"2007.02195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2020-07-04T22:07:04Z","cross_cats_sorted":["nlin.CD","physics.data-an"],"title_canon_sha256":"5cf33354814db3d86bc77a9b440011fef650d085951c507083214d88fee7217f","abstract_canon_sha256":"b802b8f7ccdae3eed29b70b610d540c74d1869dbcc694c1c5bd12d637b6a8f9e"},"schema_version":"1.0"},"canonical_sha256":"adb144c264e2ce0597b835a74628f0a1511e9c484032b80f1578947ff95c4652","source":{"kind":"arxiv","id":"2007.02195","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2007.02195","created_at":"2026-07-05T01:46:22Z"},{"alias_kind":"arxiv_version","alias_value":"2007.02195v2","created_at":"2026-07-05T01:46:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.02195","created_at":"2026-07-05T01:46:22Z"},{"alias_kind":"pith_short_12","alias_value":"VWYUJQTE4LHA","created_at":"2026-07-05T01:46:22Z"},{"alias_kind":"pith_short_16","alias_value":"VWYUJQTE4LHALF5Y","created_at":"2026-07-05T01:46:22Z"},{"alias_kind":"pith_short_8","alias_value":"VWYUJQTE","created_at":"2026-07-05T01:46:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2020:VWYUJQTE4LHALF5YGWTUMKHQUF","target":"record","payload":{"canonical_record":{"source":{"id":"2007.02195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2020-07-04T22:07:04Z","cross_cats_sorted":["nlin.CD","physics.data-an"],"title_canon_sha256":"5cf33354814db3d86bc77a9b440011fef650d085951c507083214d88fee7217f","abstract_canon_sha256":"b802b8f7ccdae3eed29b70b610d540c74d1869dbcc694c1c5bd12d637b6a8f9e"},"schema_version":"1.0"},"canonical_sha256":"adb144c264e2ce0597b835a74628f0a1511e9c484032b80f1578947ff95c4652","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:46:22.824453Z","signature_b64":"724i9QJmgoWfNvGHbvEM1siLb72bQuKQiVOTbOoY90Z5vsFb8nhH8T7ipbGNTLA9j67BjOemRrN2+YsnNb/2AQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"adb144c264e2ce0597b835a74628f0a1511e9c484032b80f1578947ff95c4652","last_reissued_at":"2026-07-05T01:46:22.823985Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:46:22.823985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2007.02195","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":"2026-07-05T01:46:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vr7KN/pHRiUXLznOosQnYmBRwBziNgfrVvITpN1WpKmfcOGO1hT07aEorGdHChH1RruTwyqNRrCar6R5SKhzAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T18:25:11.496053Z"},"content_sha256":"4145a38b66ed35a2f7eb12c7d3b7ce4daf40a1610edc85e22a0c91142e1bb357","schema_version":"1.0","event_id":"sha256:4145a38b66ed35a2f7eb12c7d3b7ce4daf40a1610edc85e22a0c91142e1bb357"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2020:VWYUJQTE4LHALF5YGWTUMKHQUF","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Delay-coordinate maps, coherence, and approximate spectra of evolution operators","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["nlin.CD","physics.data-an"],"primary_cat":"math.DS","authors_text":"Dimitrios Giannakis","submitted_at":"2020-07-04T22:07:04Z","abstract_excerpt":"The problem of data-driven identification of coherent observables of measure-preserving, ergodic dynamical systems is studied using kernel integral operator techniques. An approach is proposed whereby complex-valued observables with approximately cyclical behavior are constructed from a pair eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are $\\epsilon$-approximate eigenfunctions of the Koopman evolution operator of the system, with a bound $\\epsilon$ controlled by the length of the delay-embedding window, the evolution time, and"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.02195","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/2007.02195/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":"2026-07-05T01:46:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2BpLzSKFgEhfnnX0MXyQMRifaMYSL1WxVlKion7sm71+nMHLHOrQNN9xUQPsa8XCT52oGMCxshBdEa+mgjg+Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-21T18:25:11.496642Z"},"content_sha256":"97ef5ede4ad89d7d1edfdb089af7c1b56e07cb4e0db929975bbacdafe6e44599","schema_version":"1.0","event_id":"sha256:97ef5ede4ad89d7d1edfdb089af7c1b56e07cb4e0db929975bbacdafe6e44599"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/VWYUJQTE4LHALF5YGWTUMKHQUF/bundle.json","state_url":"https://pith.science/pith/VWYUJQTE4LHALF5YGWTUMKHQUF/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/VWYUJQTE4LHALF5YGWTUMKHQUF/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":"2026-08-21T18:25:11Z","links":{"resolver":"https://pith.science/pith/VWYUJQTE4LHALF5YGWTUMKHQUF","bundle":"https://pith.science/pith/VWYUJQTE4LHALF5YGWTUMKHQUF/bundle.json","state":"https://pith.science/pith/VWYUJQTE4LHALF5YGWTUMKHQUF/state.json","well_known_bundle":"https://pith.science/.well-known/pith/VWYUJQTE4LHALF5YGWTUMKHQUF/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2020:VWYUJQTE4LHALF5YGWTUMKHQUF","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":"b802b8f7ccdae3eed29b70b610d540c74d1869dbcc694c1c5bd12d637b6a8f9e","cross_cats_sorted":["nlin.CD","physics.data-an"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2020-07-04T22:07:04Z","title_canon_sha256":"5cf33354814db3d86bc77a9b440011fef650d085951c507083214d88fee7217f"},"schema_version":"1.0","source":{"id":"2007.02195","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2007.02195","created_at":"2026-07-05T01:46:22Z"},{"alias_kind":"arxiv_version","alias_value":"2007.02195v2","created_at":"2026-07-05T01:46:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2007.02195","created_at":"2026-07-05T01:46:22Z"},{"alias_kind":"pith_short_12","alias_value":"VWYUJQTE4LHA","created_at":"2026-07-05T01:46:22Z"},{"alias_kind":"pith_short_16","alias_value":"VWYUJQTE4LHALF5Y","created_at":"2026-07-05T01:46:22Z"},{"alias_kind":"pith_short_8","alias_value":"VWYUJQTE","created_at":"2026-07-05T01:46:22Z"}],"graph_snapshots":[{"event_id":"sha256:97ef5ede4ad89d7d1edfdb089af7c1b56e07cb4e0db929975bbacdafe6e44599","target":"graph","created_at":"2026-07-05T01:46:22Z","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/2007.02195/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The problem of data-driven identification of coherent observables of measure-preserving, ergodic dynamical systems is studied using kernel integral operator techniques. An approach is proposed whereby complex-valued observables with approximately cyclical behavior are constructed from a pair eigenfunctions of integral operators built from delay-coordinate mapped data. It is shown that these observables are $\\epsilon$-approximate eigenfunctions of the Koopman evolution operator of the system, with a bound $\\epsilon$ controlled by the length of the delay-embedding window, the evolution time, and","authors_text":"Dimitrios Giannakis","cross_cats":["nlin.CD","physics.data-an"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2020-07-04T22:07:04Z","title":"Delay-coordinate maps, coherence, and approximate spectra of evolution operators"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2007.02195","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":"sha256:4145a38b66ed35a2f7eb12c7d3b7ce4daf40a1610edc85e22a0c91142e1bb357","target":"record","created_at":"2026-07-05T01:46:22Z","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":"b802b8f7ccdae3eed29b70b610d540c74d1869dbcc694c1c5bd12d637b6a8f9e","cross_cats_sorted":["nlin.CD","physics.data-an"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2020-07-04T22:07:04Z","title_canon_sha256":"5cf33354814db3d86bc77a9b440011fef650d085951c507083214d88fee7217f"},"schema_version":"1.0","source":{"id":"2007.02195","kind":"arxiv","version":2}},"canonical_sha256":"adb144c264e2ce0597b835a74628f0a1511e9c484032b80f1578947ff95c4652","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"adb144c264e2ce0597b835a74628f0a1511e9c484032b80f1578947ff95c4652","first_computed_at":"2026-07-05T01:46:22.823985Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:46:22.823985Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"724i9QJmgoWfNvGHbvEM1siLb72bQuKQiVOTbOoY90Z5vsFb8nhH8T7ipbGNTLA9j67BjOemRrN2+YsnNb/2AQ==","signature_status":"signed_v1","signed_at":"2026-07-05T01:46:22.824453Z","signed_message":"canonical_sha256_bytes"},"source_id":"2007.02195","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4145a38b66ed35a2f7eb12c7d3b7ce4daf40a1610edc85e22a0c91142e1bb357","sha256:97ef5ede4ad89d7d1edfdb089af7c1b56e07cb4e0db929975bbacdafe6e44599"],"state_sha256":"063f890084664ec9b0fc40c6c86e1cd186f72695b63d97ef077b473d89c751d2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Qs5VF/aCjA3r+m5Ph3aYtq9sAcp0ZYbspP/M/bBR9d9G6xiwh5cZLtJg4bH1CW+Yq2u4NK+ASnBHEhtJB7qKDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-21T18:25:11.502943Z","bundle_sha256":"f90c626e1ac2d6247b31ed3f0c99277751a234637c5565e8ae8bf1740b0d46b3"}}