{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2007:ABC4FGBJO4LWIAGTJ3D2LYNTNZ","short_pith_number":"pith:ABC4FGBJ","schema_version":"1.0","canonical_sha256":"0045c2982977176400d34ec7a5e1b36e780de8ce94929292bad419895c51cf77","source":{"kind":"arxiv","id":"math/0702322","version":2},"attestation_state":"computed","paper":{"title":"Proper actions and proper invariant metrics","license":"","headline":"","cross_cats":["math.GN"],"primary_cat":"math.MG","authors_text":"Antonios Manoussos, Gennady Noskov, Herbert Abels","submitted_at":"2007-02-12T11:51:00Z","abstract_excerpt":"We show that if a (locally compact) group $G$ acts properly on a locally compact $\\sigma$-compact space $X$ then there is a family of $G$-invariant proper continuous finite-valued pseudometrics which induces the topology of $X$. If $X$ is furthermore metrizable then $G$ acts properly on $X$ if and only if there exists a $G$-invariant proper compatible metric on $X$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0702322","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.MG","submitted_at":"2007-02-12T11:51:00Z","cross_cats_sorted":["math.GN"],"title_canon_sha256":"298820f0e02a0504636c0eed6b6d9a38b86f02db332cc86d09d691ff21933b98","abstract_canon_sha256":"9bd638bb8d6bf14cf2e695085f48f8f93e9609ebd9342a8925fabad7fb17d7e6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:45.278000Z","signature_b64":"+7mRx9UCQ4y/faeUEazTD29eUwnth+MKn0l+wlccy9J9lSm1oqNwbHMhfl4KMiZNBfF2GMI1xs90U/efse3SDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0045c2982977176400d34ec7a5e1b36e780de8ce94929292bad419895c51cf77","last_reissued_at":"2026-05-18T02:57:45.277610Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:45.277610Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proper actions and proper invariant metrics","license":"","headline":"","cross_cats":["math.GN"],"primary_cat":"math.MG","authors_text":"Antonios Manoussos, Gennady Noskov, Herbert Abels","submitted_at":"2007-02-12T11:51:00Z","abstract_excerpt":"We show that if a (locally compact) group $G$ acts properly on a locally compact $\\sigma$-compact space $X$ then there is a family of $G$-invariant proper continuous finite-valued pseudometrics which induces the topology of $X$. If $X$ is furthermore metrizable then $G$ acts properly on $X$ if and only if there exists a $G$-invariant proper compatible metric on $X$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0702322","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":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0702322","created_at":"2026-05-18T02:57:45.277672+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0702322v2","created_at":"2026-05-18T02:57:45.277672+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0702322","created_at":"2026-05-18T02:57:45.277672+00:00"},{"alias_kind":"pith_short_12","alias_value":"ABC4FGBJO4LW","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_16","alias_value":"ABC4FGBJO4LWIAGT","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_8","alias_value":"ABC4FGBJ","created_at":"2026-05-18T12:25:55.427421+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ABC4FGBJO4LWIAGTJ3D2LYNTNZ","json":"https://pith.science/pith/ABC4FGBJO4LWIAGTJ3D2LYNTNZ.json","graph_json":"https://pith.science/api/pith-number/ABC4FGBJO4LWIAGTJ3D2LYNTNZ/graph.json","events_json":"https://pith.science/api/pith-number/ABC4FGBJO4LWIAGTJ3D2LYNTNZ/events.json","paper":"https://pith.science/paper/ABC4FGBJ"},"agent_actions":{"view_html":"https://pith.science/pith/ABC4FGBJO4LWIAGTJ3D2LYNTNZ","download_json":"https://pith.science/pith/ABC4FGBJO4LWIAGTJ3D2LYNTNZ.json","view_paper":"https://pith.science/paper/ABC4FGBJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0702322&json=true","fetch_graph":"https://pith.science/api/pith-number/ABC4FGBJO4LWIAGTJ3D2LYNTNZ/graph.json","fetch_events":"https://pith.science/api/pith-number/ABC4FGBJO4LWIAGTJ3D2LYNTNZ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ABC4FGBJO4LWIAGTJ3D2LYNTNZ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ABC4FGBJO4LWIAGTJ3D2LYNTNZ/action/storage_attestation","attest_author":"https://pith.science/pith/ABC4FGBJO4LWIAGTJ3D2LYNTNZ/action/author_attestation","sign_citation":"https://pith.science/pith/ABC4FGBJO4LWIAGTJ3D2LYNTNZ/action/citation_signature","submit_replication":"https://pith.science/pith/ABC4FGBJO4LWIAGTJ3D2LYNTNZ/action/replication_record"}},"created_at":"2026-05-18T02:57:45.277672+00:00","updated_at":"2026-05-18T02:57:45.277672+00:00"}