{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:KZJMIJJ6ESIYOW5F5WJGZEPLMZ","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":"597b1926f05a63d93dddf5f3929fae6a671af9080ea1b830dc0689aed0ea4109","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-14T11:24:52Z","title_canon_sha256":"2a657566ea3fc068e078f92bb908aa2319a3b65fadcf73b032ce668a13b7878f"},"schema_version":"1.0","source":{"id":"1109.3051","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.3051","created_at":"2026-05-18T04:13:06Z"},{"alias_kind":"arxiv_version","alias_value":"1109.3051v1","created_at":"2026-05-18T04:13:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.3051","created_at":"2026-05-18T04:13:06Z"},{"alias_kind":"pith_short_12","alias_value":"KZJMIJJ6ESIY","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"KZJMIJJ6ESIYOW5F","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"KZJMIJJ6","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:7cbb19d4337077da30617b2301d3db97786b63b9a28f24f8e171e0ae0338c7fc","target":"graph","created_at":"2026-05-18T04:13:06Z","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 give necessary and sufficient conditions for both square integrability and smoothness for densities of a probability measure on a compact connected Lie group.","authors_text":"David Applebaum","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-14T11:24:52Z","title":"Smoothness of Densities on Compact Lie Groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.3051","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:d0503e096f98ef6eef63b1d40dac65e6562c9f44d452053b7a1d9efe5a94679b","target":"record","created_at":"2026-05-18T04:13:06Z","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":"597b1926f05a63d93dddf5f3929fae6a671af9080ea1b830dc0689aed0ea4109","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-09-14T11:24:52Z","title_canon_sha256":"2a657566ea3fc068e078f92bb908aa2319a3b65fadcf73b032ce668a13b7878f"},"schema_version":"1.0","source":{"id":"1109.3051","kind":"arxiv","version":1}},"canonical_sha256":"5652c4253e2491875ba5ed926c91eb667580b5c6afcedba976d6ff2949e7c7f3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5652c4253e2491875ba5ed926c91eb667580b5c6afcedba976d6ff2949e7c7f3","first_computed_at":"2026-05-18T04:13:06.398849Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:13:06.398849Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PqxfbIW+9dYVpX8phl3WIuxy+9hChOXdpDaeTg335Q6VhuDyemTisxmpGxiXhmUQbB9PF+sYIsNh+cecBXzTCA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:13:06.399420Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.3051","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d0503e096f98ef6eef63b1d40dac65e6562c9f44d452053b7a1d9efe5a94679b","sha256:7cbb19d4337077da30617b2301d3db97786b63b9a28f24f8e171e0ae0338c7fc"],"state_sha256":"a9d3d3975199cb9794a893fa2c78900d5acbbaddf5c9835d783d72a005e436c4"}