{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:MOJT3WGDHSF2PHH5I5Y54ID22A","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":"4776d0d7bfcffcb4f8a130152f8f1333738ae115e69361ccc2a92ae58e73df8f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-02T21:49:46Z","title_canon_sha256":"c5dc87a61e1db5eae676f922238763730ac548ec068e94908543631a6f581292"},"schema_version":"1.0","source":{"id":"2606.04245","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.04245","created_at":"2026-06-04T01:09:00Z"},{"alias_kind":"arxiv_version","alias_value":"2606.04245v1","created_at":"2026-06-04T01:09:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.04245","created_at":"2026-06-04T01:09:00Z"},{"alias_kind":"pith_short_12","alias_value":"MOJT3WGDHSF2","created_at":"2026-06-04T01:09:00Z"},{"alias_kind":"pith_short_16","alias_value":"MOJT3WGDHSF2PHH5","created_at":"2026-06-04T01:09:00Z"},{"alias_kind":"pith_short_8","alias_value":"MOJT3WGD","created_at":"2026-06-04T01:09:00Z"}],"graph_snapshots":[{"event_id":"sha256:72e725f85bcf1c981c2eb3575b23118f96455013fe8494eb2ec8486dfc9fa774","target":"graph","created_at":"2026-06-04T01:09:00Z","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/2606.04245/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This article investigates the structural properties of stochastic processes relative to a generalized single jump filtration, extending the framework introduced by A.A. Gushchin (2020) to the case of a non-trivial initial $\\sigma$-algebra $\\mathscr{H}$. By leveraging the general theory of processes and optional projection techniques, we establish fundamental measurability criteria for random variables and a complete characterization of stopping times and adapted processes. Furthermore, we derive comprehensive martingale and local martingale criteria, providing necessary and sufficient conditio","authors_text":"Assylliya K. Zhunussova (Lomonosov Moscow State University, Moscow, Russia)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-02T21:49:46Z","title":"Properties of a Special Type of Filtration and its Martingale Criteria"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.04245","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:fa3084954ef25ac021a81a887424c5f89568a021f0c7237ed0512cbd42bd23f1","target":"record","created_at":"2026-06-04T01:09:00Z","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":"4776d0d7bfcffcb4f8a130152f8f1333738ae115e69361ccc2a92ae58e73df8f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-02T21:49:46Z","title_canon_sha256":"c5dc87a61e1db5eae676f922238763730ac548ec068e94908543631a6f581292"},"schema_version":"1.0","source":{"id":"2606.04245","kind":"arxiv","version":1}},"canonical_sha256":"63933dd8c33c8ba79cfd4771de207ad02e03062ade0adad543e148a05d8b00bc","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"63933dd8c33c8ba79cfd4771de207ad02e03062ade0adad543e148a05d8b00bc","first_computed_at":"2026-06-04T01:09:00.227860Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-04T01:09:00.227860Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/mwcpYDVe64nJ2CVhQLNJIidB+T+XEqDgNm+bC0S9lAAnf3T8jv9B51HDzFxkrRTQOmWS388KizYA4gnQMVRDg==","signature_status":"signed_v1","signed_at":"2026-06-04T01:09:00.228520Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.04245","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fa3084954ef25ac021a81a887424c5f89568a021f0c7237ed0512cbd42bd23f1","sha256:72e725f85bcf1c981c2eb3575b23118f96455013fe8494eb2ec8486dfc9fa774"],"state_sha256":"cef1245727f5a85275f6605e1e6d02f0ba0f13aa7f4bfae027f7d771e27a353f"}