{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:BM54VWEUTCEQ5KDLMGXN5ADSUT","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":"a5d36cc8098cc8d1ceabb467353aa9ca729c8dde55bb419e6cd5566b366accfa","cross_cats_sorted":["math.CO","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2014-12-08T20:51:36Z","title_canon_sha256":"e7f0f74301664ec79f4109c7353da6b520e412f2cedeec033115640bcabfac37"},"schema_version":"1.0","source":{"id":"1412.2735","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.2735","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"arxiv_version","alias_value":"1412.2735v3","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.2735","created_at":"2026-05-18T01:01:29Z"},{"alias_kind":"pith_short_12","alias_value":"BM54VWEUTCEQ","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_16","alias_value":"BM54VWEUTCEQ5KDL","created_at":"2026-05-18T12:28:22Z"},{"alias_kind":"pith_short_8","alias_value":"BM54VWEU","created_at":"2026-05-18T12:28:22Z"}],"graph_snapshots":[{"event_id":"sha256:2e84c6b57643c3a7662e34d51fd1c2d4b816186fb24c307b6507dfd008b951bc","target":"graph","created_at":"2026-05-18T01:01:29Z","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 consider the space of countable structures with fixed underlying set in a given countable language. We show that the number of ergodic probability measures on this space that are $S_\\infty$-invariant and concentrated on a single isomorphism class must be zero, or one, or continuum. Further, such an isomorphism class admits a unique $S_\\infty$-invariant probability measure precisely when the structure is highly homogeneous; by a result of Peter J. Cameron, these are the structures that are interdefinable with one of the five reducts of the rational linear order $(\\mathbb{Q}, <)$.","authors_text":"Aleksandra Kwiatkowska, Cameron Freer, Nathanael Ackerman, Rehana Patel","cross_cats":["math.CO","math.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2014-12-08T20:51:36Z","title":"A classification of orbits admitting a unique invariant measure"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.2735","kind":"arxiv","version":3},"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":"sha256:683495427a946ca9a90242f84da19ca09f4a80177b245648fc8969ebace5871a","target":"record","created_at":"2026-05-18T01:01:29Z","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":"a5d36cc8098cc8d1ceabb467353aa9ca729c8dde55bb419e6cd5566b366accfa","cross_cats_sorted":["math.CO","math.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.LO","submitted_at":"2014-12-08T20:51:36Z","title_canon_sha256":"e7f0f74301664ec79f4109c7353da6b520e412f2cedeec033115640bcabfac37"},"schema_version":"1.0","source":{"id":"1412.2735","kind":"arxiv","version":3}},"canonical_sha256":"0b3bcad89498890ea86b61aede8072a4c01a74048b2209c9ccde645c08873d12","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0b3bcad89498890ea86b61aede8072a4c01a74048b2209c9ccde645c08873d12","first_computed_at":"2026-05-18T01:01:29.403929Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:01:29.403929Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"M+UeSZ+xfFm4GRE0+i2+cWbNZWL3gIEirzCcmUtQoDMd/F8Mi/bDDQ4HAuQEzT0Qd/nz57KzaizZXkXk2vtBDw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:01:29.404345Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.2735","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:683495427a946ca9a90242f84da19ca09f4a80177b245648fc8969ebace5871a","sha256:2e84c6b57643c3a7662e34d51fd1c2d4b816186fb24c307b6507dfd008b951bc"],"state_sha256":"f9b27c359f1bfaf02373c754a3a6963a78c2c667c11824b720014401b4cbc845"}