{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:74JIQRI3FURVFN235TVWKUAB3Q","merge_version":"pith-open-graph-merge-v1","event_count":3,"valid_event_count":3,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f693e8b52cddd1bf9968137caebef4f765ea9afe55edafc09ceee9c077cb6cc3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-22T08:12:38Z","title_canon_sha256":"a88950773635def4225c0b10ac0998467f8ea9b3d8915c13ff7a03c34be21b68"},"schema_version":"1.0","source":{"id":"2605.23349","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2605.23349","created_at":"2026-05-25T02:01:50Z"},{"alias_kind":"arxiv_version","alias_value":"2605.23349v1","created_at":"2026-05-25T02:01:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.23349","created_at":"2026-05-25T02:01:50Z"},{"alias_kind":"pith_short_12","alias_value":"74JIQRI3FURV","created_at":"2026-05-25T02:01:50Z"},{"alias_kind":"pith_short_16","alias_value":"74JIQRI3FURVFN23","created_at":"2026-05-25T02:01:50Z"},{"alias_kind":"pith_short_8","alias_value":"74JIQRI3","created_at":"2026-05-25T02:01:50Z"}],"graph_snapshots":[{"event_id":"sha256:3145a7a477119e769de8bbcb623ebc9c235af82daabdceda25c0a15bcfb92086","target":"graph","created_at":"2026-05-25T02:01:50Z","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/2605.23349/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Joinings are fundamental global objects in ergodic theory, yet in compact metric models one naturally observes only finite orbit-distance patterns. We bridge this gap by introducing multi-particle distance arrays, which sample finite orbit segments and record their joint metric evolution. In the anchored fixed-model setting, this framework yields a purely finite-observable characterization of Furstenberg disjointness: two systems are disjoint if and only if all their anchored multi-orbit distance-array projections are independent. The structural engine behind this criterion is a marked and col","authors_text":"Ao Xu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-22T08:12:38Z","title":"Observing Joinings: A Distance-Array Characterization of Furstenberg Disjointness"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.23349","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:f23a62f182da06085bf032b0c6bec16aaacbf10b6efbb37dde3453d2c0c0e6e5","target":"record","created_at":"2026-05-25T02:01:50Z","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":"f693e8b52cddd1bf9968137caebef4f765ea9afe55edafc09ceee9c077cb6cc3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2026-05-22T08:12:38Z","title_canon_sha256":"a88950773635def4225c0b10ac0998467f8ea9b3d8915c13ff7a03c34be21b68"},"schema_version":"1.0","source":{"id":"2605.23349","kind":"arxiv","version":1}},"canonical_sha256":"ff1288451b2d2352b75beceb655001dc31383970fe0d0c5024916927becc8eb9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ff1288451b2d2352b75beceb655001dc31383970fe0d0c5024916927becc8eb9","first_computed_at":"2026-05-25T02:01:50.257161Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-25T02:01:50.257161Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"I3GNKcOxuybtgHlorAlUvALFz5OzY78lNzBc3+UW5YqvM3jg8TBsewWx4M7QkgEYKv75J0gPeKCn2YWKXmBZCQ==","signature_status":"signed_v1","signed_at":"2026-05-25T02:01:50.257804Z","signed_message":"canonical_sha256_bytes"},"source_id":"2605.23349","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f23a62f182da06085bf032b0c6bec16aaacbf10b6efbb37dde3453d2c0c0e6e5","sha256:3145a7a477119e769de8bbcb623ebc9c235af82daabdceda25c0a15bcfb92086","sha256:c0e44359f1bf729fab3855feb5fad917904de4fb359e2a4256a627d375bf4929"],"state_sha256":"42127d1880019fcf27cd1289ba6c7c55555a482161667536b60d1227ecdc796f"}