{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:FCMA7H4ACUCLA57QJBOHXXN2ZO","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":"b3e97539e78eadf02d5e654bff4d1328567ccdd2daf6d7b2ac95a09cfd2ae085","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-01-10T21:22:08Z","title_canon_sha256":"6850a64a893317c1466774448b6e5ced83991d5674ff2e1e34b57f87edb5ca28"},"schema_version":"1.0","source":{"id":"1901.04317","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1901.04317","created_at":"2026-05-17T23:56:25Z"},{"alias_kind":"arxiv_version","alias_value":"1901.04317v1","created_at":"2026-05-17T23:56:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1901.04317","created_at":"2026-05-17T23:56:25Z"},{"alias_kind":"pith_short_12","alias_value":"FCMA7H4ACUCL","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_16","alias_value":"FCMA7H4ACUCLA57Q","created_at":"2026-05-18T12:33:15Z"},{"alias_kind":"pith_short_8","alias_value":"FCMA7H4A","created_at":"2026-05-18T12:33:15Z"}],"graph_snapshots":[{"event_id":"sha256:9f4a3f0436badc1985a0c88080db537c920909f3f5339b8344c84d229216847d","target":"graph","created_at":"2026-05-17T23:56:25Z","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":"Let $\\Omega$ be a metric space, $A^t$ denote the metric neighborhood of the set $A\\subset\\Omega$ of the radius $t$; ${\\mathfrak O}$ be the lattice of open sets in $\\Omega$ with the partial order $\\subseteq$ and the order convergence. The lattice of $\\mathfrak O$-valued functions of $t\\in(0,\\infty)$ with the point-wise partial order and convergence contains the family ${I\\mathfrak O}=\\{A(\\cdot)\\,|\\,\\,A(t)=A^t,\\,\\,A\\in{\\mathfrak O}\\}$. Let $\\widetilde\\Omega$ be the set of atoms of the order closure $\\overline{I\\mathfrak O}$. We describe a class of spaces for which the set $\\widetilde\\Omega$, equ","authors_text":"M. I. Belishev, S. A. Simonov","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-01-10T21:22:08Z","title":"The wave model of metric spaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.04317","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":"sha256:19185cce38b4fc0e261d9f813e81cb9863244b53f6c32ad08317ab4fa3544c6d","target":"record","created_at":"2026-05-17T23:56:25Z","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":"b3e97539e78eadf02d5e654bff4d1328567ccdd2daf6d7b2ac95a09cfd2ae085","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2019-01-10T21:22:08Z","title_canon_sha256":"6850a64a893317c1466774448b6e5ced83991d5674ff2e1e34b57f87edb5ca28"},"schema_version":"1.0","source":{"id":"1901.04317","kind":"arxiv","version":1}},"canonical_sha256":"28980f9f801504b077f0485c7bddbacb8eb568ea3f68cf9f416a7dd52b13883c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"28980f9f801504b077f0485c7bddbacb8eb568ea3f68cf9f416a7dd52b13883c","first_computed_at":"2026-05-17T23:56:25.849715Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:56:25.849715Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wydyQlbaWVH22uf0UVZyT0Auv3FE2JZQGrHxPOWKuEVdNcLpIl0Pkg3SUYqboocKGTAsDmkDO5BSLWhB6OL+Ag==","signature_status":"signed_v1","signed_at":"2026-05-17T23:56:25.850138Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.04317","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:19185cce38b4fc0e261d9f813e81cb9863244b53f6c32ad08317ab4fa3544c6d","sha256:9f4a3f0436badc1985a0c88080db537c920909f3f5339b8344c84d229216847d"],"state_sha256":"c0f673a1e6a87f1e3c15b291315243855689300e6a947663fe0a72320546e834"}